ASTRONOMY  DEFT 


PERIODIC  ORBITS 


HY 


F.  R.  MOULTON 


IN    COLLABOKATION    WITH 


DANIEL  BUCHANAN,  THOMAS  BUCK,  FRANK  L.  GRIFFIN, 
WILLIAM  R.  LONGLEY  AND  WILLIAM  D.  MxcMILLAN 


PUBLISHED  BY  THE  CARNEGIE  INSTITUTION  OP  WASHINGTON 
WASHINGTON,  1920 


M  B  2  6-5- 
M<. 


•    ••        i/tt. 

CARNEGIE  INSTlftlTION  OF  WASHINGTON 

/  ;;/':>:P^BLWTi-6rN'No.  161 


ASTRONOMY 


PRESS  OF  GIBSON  BROTHERS,  INC. 
WASHINGTON,  It.  C. 


INTRODUCTION. 

Tlir  problem  of  three  bodies  received  a  great  impetus  ill  1S7S,  \\hen 
Hill  published  his  celebrated  researches  upon  the  lunar  theory.  His  inves- 
tigations \\ere  carried  out  with  practical  objects  in  mind,  and  compara- 
tively little  attention  was  given  io  ihe  underlying  logic  of  the  processes 
which  he  invented.  For  example,  the  legitimacy  of  the  use  of  infinite 
determinants  was  assumed,  the  validity  of  the  solution  of  infinite  systems  of 
non-linear  equations  was  not  questioned,  and  the  conditions  for  the  conver- 
gence of  the  infinite  series  which  he  used  were  stated  to  be  quite  unknown. 
These  deficiencies  in  the  logic  of  his  work  do  not  detract  from  the  brilliancy 
and  value  of  his  ideas,  and  his  skill  in  carrying  them  out  excites  only  the 
highest  admiration. 

The  work  of  Hill  was  followed  in  the  early  nineties  by  the  epoch-making 
•  ('  Poincare.  which  were  published  in  detail  in  his  Les  Mithodes 
Xiiurfllfx  ill  In  MI'CII >u'i/m  ('«'/•*/».  Poincar<5  brought  to  boar  on  the  prob- 
lem all  the  resources  of  modern  analysis.  The  new  methods  of  treating  the 
difficult  problem  of  three  bodies  which  he  invented  were  so  numerous  and 
powerful  as  to  be  positively  bewildering.  They  opened  so  many  new  fields 
that  a  generation  will  lx>  required  for  their  complete  exploration.  On  the 
one  hand,  the  results  were  in  the  direction  of  purely  theoretical  considera- 
tions, in  which  Birkoff  has  recently  made  noteworthy  extensions;  on  the 
other  hand,  they  foreshadowed  somewhat  dimly  methods  which  will  doubt- 
less be  of  great  importance  in  practical  applications  in  celestial  mechanics. 
The  researches  of  Poincare  are  scarcely  less  revolutionary  in  character  than 
wen-  those  of  Newton  when  he  discovered  the  law  of  gravitation  and  laid 
the  foundations  of  celestial  mechanics. 

In  1890  Sir  ( Icorge  Darwin  published  an  extensive  paper  on  the  problem 
of  three  bodies  in  Adu  MnUu-mntica.  In  mathematical  spirit  it  was  similar 
to  the  work  of  Hill;  indeed,  the  methods  used  were  essentially  those  of  Hill, 
but  the  problem  treated  was  considerably  different.  For  a  ratio  of  the  finite 
masses  of  ten  to  one,  Darwin  undertook  to  discover  by  numerical  processes 
all  the  j>eriodic  orbits  of  certain  types  and  to  follow  their  changes  with 
\  arying  values  of  the  Jacobian  constant  of  integration.  This  program  was 
excellently  carried  out  at  the  cost  of  a  great  amount  of  labor.  It  gave 
specific  numerical  results  for  many  orbits  in  a  particular  example. 

The  investigations  contained  in  this  volume  were  begun  in  1900  and. 
with  the  exception  of  the  last  chapter,  they  were  completed  by  1912.  Those 
not  made  by  myself  were  carried  out  by  students  who  made  their  doctorates 
under  my  direction. 

in 

r/39883 


iv  INTRODUCTION. 

The  following  chapters  have  been  heretofore  published  in  substance : 

I.  Sections  III  and  IV     American  Journal  of  Mathematics,  vol.  xxxm  (1911). 

II.  Astronomical  Journal,  vol.  xxv  (1907). 

III.  Rendiconti  Matematico  di  Palermo,  vol.  xxxn  (1911). 

IV.  Transactions  of  the  American  Mathematical  Society,  vol.  xi  (1910). 
VII.  Mathematische  Annalen,  vol.  LXXIII  (1913). 

VIII.  Annals  of  Mathematics,  2d  Series,  vol.  12  (1910) 

XI.  Transactions  of  the  American  Mathematical  Society,  vol.  vn  (190(i). 

XII.  Transactions  of  the  American  Mathematical  Society,  vol.  xni  (1912). 

XIII.  Transactions  of  the  American  Mathematical  Society,  vol.  vin  (1907). 

XIV.  Transactions  of  the  American  Mathematical  Society,  vol.  ix  (1908). 
XV.  Proceedings  of  the  London  Mathematical  Society,  Series  II,  vol.  2  (1912). 

The  investigations  and  computations  contained  in  the  last  chapter 
were  completed  in  1917. 

It  was  originally  intended  to  publish  only  the  first  fifteen  chapters, 
and  if  that  program  had  been  carried  out  they  would  have  appeared  in  1912. 
But  as  the  work  of  printing  progressed  the  ideas  contained  in  the  last  chapter 
were  being  developed  and  the  computations  were  begun.  It  was  thought 
that  an  even  more  nearly  complete  and  certain  idea  of  the  evolution  of  peri- 
odic orbits  with  changing  parameters  could  be  obtained  in  a  year  than  were 
obtained  in  five  years.  The  difficulties  and  enormous  amount  of  labor 
involved  were  not  foreseen.  No  one  can  now  read  with  better  appreciation 
than  I  the  following  words  from  Darwin's  introduction  to  his  paper : 

"As  far  as  I  can  see,  the  search  resolves  itself  into  the  discussion  of  particular 
cases  by  numerical  processes,  and  such  a  search  necessarily  involves  a  prodigious 
amount  of  work.  It  is  not  for  me  to  say  whether  the  enormous  amount  of  labor  I 
have  undertaken  was  justifiable  in  the  first  instance ;  but  I  may  remark  that  I  have 
been  led  on  by  the  interest  of  my  results,  step  by  step,  to  investigate  more,  and 
again  more,  cases." 

The  results  which  now  appear  had  all  been  obtained  when  service  in 
the  army  made  it  necessary  to  lay  them  aside  before  the  final  chapter  could 
be  put  into  form  for  publication.  After  they  had  been  laid  aside  for  about 
two  years  it  was  not  easy  to  gather  up  the  details  again  and  to  arrange  them 
in  a  systematic  order.  This  explains  the  long  delay  in  the  appearance  of 
this  volume.  It  is  clear  that  it  was  in  no  wise  due  to  the  Carnegie  Institu- 
tion of  Washington.  Indeed,  the  patience  of  President  Woodward  with 
long  and  expensive  delays  has  been  far  beyond  what  could  reasonably  have 
been  expected. 

In  the  greater  part  of  this  work  complete  mathematical  rigor  has  been 
insisted  upon.  On  the  other  hand,  the  developments  have  been  in  a  form 
applicable  to  practical  problems  in  celestial  mechanics.  For  example, 
sections  III  and  IV  of  Chapter  I  treat  non-homogeneous  equations  of  the 
types  which  arise  in  practical  problems;  Chapter  II  is  devoted  to  questions 
which  have  long  been  classic  in  celestial  mechanics;  Chapter  III  contains, 
among  other  things,  a  new  and  rigorous  treatment  of  Hill's  differential 
equation  with  periodic  coefficients;  Chapter  IV  treats  a  problem  that  arises, 


at  least  approximately,  in  the  solar  system;  ( 'liaptcr  V  is  developed  in  a  form 
suitable  for  numerical  applications;  Chapter  VI  is  an  alternative  treatment 
of  the  same  problem,  and  in  ( 'hapter  VII  an  extension  of  the  problem 
in\olving  entirely  new  types  of  difficulties  is  found  and  more  powerful 
methods  of  treatment  are  required;  ('hapter  XI  covers  the  same  ground  as 
Hill's  work  on  the  moon's  variational  orbit  and  Brown's  work  on  the  paral- 
lactic  terms;  ( 'hapter  XII  contains  the  corresponding  discussion  for  superior 
planet-;  and  ( 'hapter  X  I  V  treats  a  problem  similar  to  that  presented  by  the 
satellite  systems  of  Jupiter  and  Saturn  or  by  the  planetary  system.  Chapter 
XV  contains  a  discussion  of  limiting  cases  of  jxriodic  orbits,  namely,  closed 
orbits  of  ejection.  It  forms  a  basis  for  part  of  the  work  of  the  last  chapter, 
and  it  may  be  found  to  have  practical  applications  in  the  escape  of  atmos- 
pheres. The  last  chapter  is  an  attempt  to  trace  out  the  evolution  of  periodic 
orbits  as  the  parameters  on  which  they  depend  are  varied.  In  spite  of  the 
fact  that  infinitely  many  families  of  periodic  orbits  were  found,  whereas  only 
a  few  such  families  were  previously  known,  the  discussion  remains  in  cer- 
tain respects  incomplete.  It  >hould  be  stated  that  many  results  were  found 
which  have  not  been  included  because  they  did  not  contribute  to  the  solu- 
tion of  the  particular  question  under  consideration.  For  example,  a  series 
of  orbits  asymptotic  to  the  collinear  equilibrium  points  was  computed. 
The  amount  of  labor  the  last  chapter  cost  can  scarcely  be  overestimated. 

I  should  not  be  true  to  my  own  feelings  if  I  did  not  express  the  appre- 
ciation  of  the  assistance  of  my  collaborators.  The  association  with  them 
has  been  a  deep  source  of  satisfaction  and  inspiration.  They  are  to  be  held 
accountable  only  for  those  chapters  which  appear  under  their  names.  Most 
of  the  computations  on  which  many  of  the  results  of  the  last  chapter 
are  based  were  made  by  Dr.  W.  L.  Hart  and  Dr.  I.  A.  Barnett.  Without 
assistance  of  such  a  lugh  order  the  laborious  computations  could  not  have 
been  carried  out. 

F.  R.  MOULTON. 


CONTENTS. 

C'llU'tKIl    I.    ('Kit  IAIN    TllK.iKKM-    o\    1\1I'I.|.I|     I  I  NOTIONS   AND    DlFFEHENTIAL 

Kqr  \IIM\-.     (H\   K.   H.   Moi  i.rox  AND  W.  D.  McMiLLAN.) 

I.  Solution  »f  Im/ilicit  Functions.  PAGE. 

1.  Formal  solution  of  simultaneous  equal  ion-;  when  (lie  functional  determinant 

i-*  distinct  from  zero  at  the  origin 1 

2.  Proof  of  convergence  of  the  solution-                2 

:J.  <  ienerali/.ation  to  many  parameters                 "i 

4.  'l'h«¥  functional  determinant  /ero  hut  not  all  its  first  minors  zero  at  origin  5 

.V  ( 'ase  where  P  (a,  M)  -a*  P,  (o,  M) ~MX  PI  (a,  M) 6 

6.  A  second  simple  case (i 

7.  ( Icneral  case  of  power  series  in  two  variables • 6 

I 1.  Solution.*  <>f  l)i]fi r< nlinl  Ki/Hiilions  as  Power  Series  in  Parameters. 

8.  The  ty |H>S  of  equations  treated 10 

9.  Formal  solution  of  differential  equations  of  Type  1 10 

10.  Determination  of  the  constants  of  integration  in  Tyjie  I 11 

1 1 .  Proof  of  the  convergence  of  the  solutions  of  Type  1 11 

12.  (iencralization  to  many  parameters 14 

1U.  ( Jenerali/ation  of  the  parameter 14 

1  I.  Formal  solution  of  differential  equations  of  Type  II 15 

!.">.  Determination  of  the  constants  of  integration  in  Type  II Hi 

lii.  Pnx»f  of  the  convergence  of  the  solutions  of  Type  II Hi 

17.  Case  of  homogeneous  linear  equations 19 

III.  Homogeneous  Linear  Differential  Equations  with  Periodic  Coefficients. 

18.  The  determinant  of  a  fundamental  set  of  solutions 21 

19.  The  character  of  the  solutions  of  a  set  of  linear  homogeneous  differential 

equations  with  uniform  periodic  coefficients 22 

20.  Solutions  when  the  roots  of  the  fundamental  equation  are  all  distinct ...  24 

21.  Solutions  when  the  fundamental  equation  has  multiple  roots I'll 

22.  The  characteristic  equation  when  the  coefficients  of  the  differential  equa- 

tions are  expansible  as  power  series  in  a  parameter  n 30 

23.  Solutions  when  a(,0),  a£",  . . .  ,  a™  are  di-tinct  and  their  differences  are 

not  congruent  to  zero  mod.  V— 1 32 

24.  Solutions  when  no  two  aj°'  are  equal  but  when  a!}"—  o|l"  is  congruent 

to  zero  mod.  V—  1 34 

25.  Solutions  when,  for  n  =  0,  the  characteristic  equation  has  a  multiple  root. .         35 

26.  Construction  of  the  solutions  when,  for  M  =  0,  the  roots  of  the  character- 

istic equation  are  di-iim-t  and  their  differences  are  not  congruent  to 

zero  mod.  V— 1 36 

27.  Construction  of   the  solutions  when  the  difference  of   two  roots  of  the 

characteristic  equation  is  congruent  to  zero  mod.  \/  —  1 40 

28.  Construction  of  the  solutions  when  two  roots  of  the  characteristic  equation 

are  equal 43 

IV.  Non-Homogeneous  Linear  Differential  Equations. 

29.  Case  where  the  right  members  are  periodic  with  the  period  2r  and  the  a, 

are  distinct 40 

30.  Case  where  the  right  members  are  periodic  terms  multiplied  Hy  an  exponen- 

tial, and  the  a,  are  distinct 47 

31.  Case  where  two  characteristic  exponents  are  equal  and  the  right  members 

are  periodic t^ 

V.  Equations  of  Variation  and  their  Characteristic  Exponents. 

32.  The  equations  of  variations 50 

33.  Theorems  on  the  characteristic  exponents ' ">1 

VII 


VIII  CONTENTS. 

CHAPTER  II.  ELLIPTIC  MOTION.  PAGE. 

34.  The  differential  equations  of  motion 55 

35.  Form  of  the  solution 

36.  Direct  construction  of  the  solution 58 

37.  Additional  properties  of  the  solution 

38.  Problem  of  the  rotating  ellipse GO 

39.  The  circular  solution 

40.  Existence  of  the  non-circular  solutions 

41.  Direct  construction  of  the  non-circular  solution 64 

42.  Properties  of  the  solution 66 

CHAPTER  III.  THE  SPHERICAL  PENDULUM. 

I.  Solution  of  the  Z-Equation. 

43.  The  differential  equations 67 

44.  Transformation  of  the  2-equation 68 

45.  First  demonstration  that  the  solution  of  (12)  is  periodic  and  that  n  and  the 

period  are  expansible  as  power  series  in  n 69 

46.  Second  demonstration  that  the  solution  of  equation  (12)  is  periodic 71 

47.  Third  proof  that  the  solution  of  (12)  is  periodic 73 

48.  Direct  construction  of  the  solution 73 

49.  Construction  of  the  solution  from  the  integral 76 

II.  Digression  on  Hill's  Equation. 

50.  The  x-equation 

51.  The  characteristic  equation 

52.  The  form  of  the  solution 81 

53.  Direct  construction  of  the  solutions  in  Case  1 83 

54.  Direct  construction  of  the  solutions  in  Case  II 85 

55.  Direct  construction  of  the  solutions  in  Case  III 87 

III.  Solution  of  the  X-  and  Y-Equations  for  the  Spherical  Pendulum. 

56.  Application  to  the  spherical  pendulum 

57.  Application  to  the  simple  pendulum 90 

58.  Application  of  the  integral  relations 93 

CHAPTER  IV.  PERIODIC  ORBITS  ABOUT  AN  OBLATE  SPHEROID. 
(By  WILLIAM  DUNCAN  MACMILLAN.) 

59.  Introduction 99 

60.  The  differential  equations 100 

61.  Surfaces  of  zero  velocity 101 

I.  Orbits  Re-entrant  after  one  Revolution. 

62.  Symmetry 102 

63.  Existence  of  periodic  orbits  in  the  equatorial  plane 103 

64.  Existence  of  periodic  orbits  which  are  inclined  to  the  equatorial  plane.  .  .  104 

65.  Existence  of  orbits  in  a  meridian  plane 106 

66.  Construction  of  periodic  solutions  in  the  equatorial  plane 107 

67.  Construction  of  periodic  solutions  for  orbits  inclined  to  the  equatorial  plane.  Ill 

68.  Construction  of  periodic  solutions  in  a  meridian  plane 114 

II.  Orbits  Re-entrant  after  many  Revolutions. 

69.  The  differential  equations 116 

70.  The  equations  of  variation 118 

71.  Special  theorems  for  the  non-homogeneous  equations 127 

72.  Integration  of  the  differential  equations 132 

73.  Existence  of  periodic  orbits  having  the  period  2i<ir 140 

74.  Construction  of  the  solutions  with  the  period  2o- 143 


CONTKNTS  IX 

CII.MTKH  V.  OSCILLATING   S  MM. 1.1  IK-    MI.. IT  TIIK  STRAIGHT-LINE  EQUI- 

i.iHitn  M    PIMM.-.       KIK-I    MKIIIOD.)  I-AOE. 

7.">.  Statement  of  problem 

7li.   'I'lie  differential  e<|iiat ions  ol'  niolioii       

77.    Regions  of  convergence  of  the  -cries  P,,  Pt,  P, !•>! 

7v    Introduction  <>f  the  parameterM  t  and  5 155 

Til.   .lacobi's  integral  156 

sn    The  -yimnetry  theorem  156 
si.  Outline  of  -tcps  for  proving  the  existence  of  |>eriodic  solutions  of  equa- 

tions      157 

sj     i  leneral  M.lulions  of  conations  (12)  for  i-«  =  0 

83.  I'erio<lic  solution-^  when  6  =  «  =  0 H><> 

84.  Noimal  form  for  the  different iiil  equations K>1 

s.'i.   1  \i-ieiice  of  |>eriodie  orbits  of  Class  A "»;f 

Stl.  Some  properties  of  solutions  of  Class  A 166 

s7.    Din-it  construction  of  the  solutions  for  Class  A 168 

88.  Additional  pro|>erties  of  orliits  of  Class  A 

89.  Application  of  .Jacolii's  integral  to  the  orbits  of  Class  A 17ii 

90.  Numerical  examples  of  orbits  of  Class  A 179 

'.M.  •  onstruction  of  a  prescribed  orbit  of  Class  A 180 

ii'J.   Kxistenee  of  orbits  of  Class  B 

!M.   Direct  construction  of  the  solutions  for  Class  B .  183 

1U.  Additional  properties  of  the  orbits  of  Class  B 

ii.V  Numerical  example  of  orbits  of  Class  B 190 

96.  On  the  existence  of  orbits  of  Class  C 191 

(  'IIAPTER  VI.  OSCILLATING  SATELLITES.     (SECOND  METHOD.) 

97.  Outline  of  method 199 

98.  The  differential  equations 199 

99.  Integration  of  the  differential  equations .  .       201 

100.  Kxistenre  of  periodic  solutions 202 

101.  Direct  construction  of  the  solutions  for  Class  A 210 

102.  Direct  construction  of  the  solutions  for  Class  B 213 

CHAPTER  VII.  <  >s<  ILLATING  SATELLITES  WHEN  THE  FINITE  MASSES 
DESCRIBE  ELLIPTICAL  ORBITS. 

103.  The  differential  equations  of  motion 

104.  The  elliptical  solution 218 

105.  Equations  for  the  oscillations 

106.  The  symmetry  theorem 

107.  Integration  of  equations  (13) 

108.  The  terms  of  the  second  degree —  

109.  The  terms  of  the  third  degree 

1 10.  General  properties  of  the  solutions 

111.  Conditions  for  the  existence  of  symmetrical  periodic  orbits 230 

112.  The  existence  of  three-dimensional  symmetrical  periodic  orbits.  .  . 

113.  Case  I,  n  even.  .V  odd -:<» 

114.  Case  II,  n  odd 

1 1 .").  Convergence 

116.  The  existence  of  two-dimensional  symmetrical  periodic  orbits 239 

117.  Case  I,  n  even 240 

118.  Case  II,  n  aid 241 


X  CONTENTS. 

CHAPTER  VII — Continued. 

Construction  of  Three-Dimensional  Periodic  Solutions. 

119.  Defining  properties  of  the  solutions 241 

120.  Coefficients  of  Xi 241 

121.  Coefficients  of  X 243 

122.  Coefficients  of  X? 245 

123.  Coefficients  of  X2 248 

124.  Coefficients  of  X? 248 

125.  The  general  step  for  the  x  and  ^-equations 250 

126.  The  general  step  for  the  z-equation 251 

Application  of  the  Integral. 

127.  Form  of  the  integral 253 

128.  The  integral  in  case  of  the  periodic  solutions 254 

129.  Determination  of  the  coefficients  of  Z%+i  when  e  =  0 257 

Determination  of  the  Coefficients  of  Z2j+i  when  e  =  0. 

130.  Case  when  e  7*  0.     General  equations  for  Z: 260 

131.  Coefficients  of  e° 262 

132.  Coefficients  of  e 263 

133.  Coefficients  of  e2  and  e4 264 

134.  General  equation  for  v  =  1 265 

135.  Terjns  independent  of  e 265 

136.  Coefficients  of  e 266 

Direct  Construction  of  the  Two-Dimensional  Symmetrical  Periodic  Solutions. 

137.  Terms  in  X* 267 

138.  Coefficients  of  X 269 

139.  Coefficients  of  X* 271 

140.  General  proof  that  the  constant  parts  of  the  right  members  of  the  first  two 

equations  of  (148)  are  identical 278 

141.  Form  of  the  periodic  solution  of  the  coefficients  of  X? 280 

142.  Coefficients  of  X2 280 

143.  Induction  to  the  general  term  of  the  solution 282 

CHAPTER  VIII.  THE  STRAIGHT-LINE  SOLUTIONS  or  THE  PROBLEM  OF  n  BODIES. 

144.  Statement  of  problem 285 

I.  Determination  of  the  Positions  when  the  Masses  are  given. 

145.  The  equations  defining  the  solutions 286 

146.  Outline  of  the  method  of  solution 288 

147.  Proof  of  theorem  (B) 289 

148.  Proof  of  theorem  (C) 290 

149.  Computation  of  the  solutions  of  equations  (6) 294 

II.  Determination  of  the  Masses  when  the  Positions  are  given. 

150.  Determination  of  the  masses  when  n  is  even 296 

151.  Determination  of  the  masses  when  n  is  odd 296 

152.  Discussion  of  case  n  =  3 297 

CHAPTER  IX.  OSCILLATING  SATELLITES  NEAR  THE  LAGRANGIAN  EQUILATERAL 
TRIANGLE  POINTS.     (By  THOMAS  BUCK.) 

153.  Introduction 299 

154.  The  differential  equations 299 

155.  The  characteristic  exponents 301 

156.  The  generating  solutions 303 

157.  General  periodicity  equations 305 


CONTENTS.  XI 

CIIAITER  IX  —  Continued.  PAGE. 

158.  The  first  generating  solution  ........................................  307 

159.  The  third  pciioniting  solution  .......................................  309 

160.  The  fourth  Kcncniting  solution  ......................................  309 

161.  Thr  fifth  generating  solution  ........................................  311 

\<\'2.  The  seventh  generating  solution  .....................................  312 

163.  (  'oust  ruction  of  the  solutions  in  the  plane  ............................  313 

164.  (  'onstriK'tion  of  the  solution  with  period  2r  ...........................  319 

(  'IIAITKU  X.  I.-HM  KI.KS-TRIANGLE  SOLUTIONS  OF  THE  PROBLEM  OK  THREE  BODIES. 

(Br  DANIEL  BUCHANAN.) 

in:.,  introdui-tiun  ......................................................  :v_>:. 


I.  /'«•/««//(•  Orbits  when  the  l-'iintt    liodiea  move  in  a  Ciri-li   <ni<l  tin    Tl<inl  /<<»/;/  in 

Infinitesimal. 

166.  The  differential  equation  of  motion  ..................................  325 

1(17.  1'roof  of  existence  of  a  periodic  solution  of  equation  (1)  ................  326 

ins    I  )irect  construction  of  the  periodic  solution  of  equation  (1)  ..............  328 

II.  ,S;/'"""  ''•«•"/  I'eriodic  Orbits  when  the  Finite  Bodies  move  in  Ellipses  <m<l  tin  Tltinl 

liixlij  is  Infinitesimal. 

Hi1.).  Tin-  differential  equation  ...........................................  330 

170.  Determination  of  the  period  hy  ft  necessary  condition  for  :i  jxriodic  solu- 

tion of  (23)    ...........  !  .....................................  331 

171.  Existence  of  symmH  rionl  periodic  orbits  ..............................  333 

172.  Direct  construction  of  symmetrical  periodic  solution-  ot    _':?»  ...........  335 

III.  Periodic  Orbits  when  the  Three  Bodies  are  Finite. 

173.  The  differential  equations  ..........................................  340 

174.  Proof  of  existence  of  periodic-  solutions  of  equations  (70)  ................  341 

1  75.  Proof  that  all  the  periodic  orbite  are  symmetrical  ......................  346 

176.  Direct  construction  of  the  periodic  solutions  of  (70)  ....................  348 

177.  The  periodic  solution  of  equation  (69)  ................................  353 

178.  The  character  of  the  periodic  solutions  ...............................  355 

(  ii  \i>i  KH  XI.  PERIODIC  ORBITS  OF  INFINITESIMAL  SATELLITES  AND  INFERIOR  PLANETS. 

179.  Introduction  ......................................................  357 

180.  The  differential  equations  ................ 

181.  Proof  of  the  existence  of  the  periodic  solutions  ........................  360 

182.  Properties  of  the  periodic  solutions  ..................................  363 

Direct  Construction  of  the  Periodic  Solutions. 

183.  General  considerations  .............................................  364 

184.  Coefficients  of  m*  ..................................................  365 

185.  Coefficients  of  m*  ..................................................  365 

186.  Coefficients  of  m4  ..................................................  366 

187.  Induction  to  the  general  step  of  the  integration  ........................  369 

188.  Application  of  Jacobi's  integral  ......................................  372 

189.  The  solutions  as  functions  of  the  Jacobian  constant  ....................  375 

190.  Applications  to  the  lunar  theory  .....................................  376 

191.  Applications  to  Darwin's  periodic  orbite  ..............................  377 


XII  CONTENTS. 

CHAPTER  XII.  PERIODIC  ORBITS  or  SUPERIOR  PLANETS. 

PAGE. 

192.  Introduction 

193.  The  differential  equation 

194.  Proof  of  the  existence  of  periodic  solutions 

195.  Practical  construction  of  the  periodic  solutions 385 

196.  Application  of  the  integral 

CHAPTER  XIII.  A  CLASS  OF  PERIODIC  ORBITS  OF  A  PARTICLE  SUBJECT  TO  THE  ATTRACTION 
OF  n  SPHERES  HAVING  PRESCRIBED  MOTION.     (Bv  W.  R.  LONGLEY.  ) 

197.  Introduction 389 

198.  Existence  of  periodic  orbits  having  no  line  of  symmetry 390 

199.  Construction  of  periodic  orbits  having  no  line  of  symmetry 396 

200.  Numerical  Example  1 402 

201 .  Some  particular  solutions  of  the  problem  of  n  bodies 404 

202.  Existence  of  symmetrical  periodic  orbits 410 

203.  Construction  of  symmetrical  periodic  orbits 414 

204.  Numerical  Example  2 415 

205.  Numerical  Example  3 417 

206.  The  undisturbed  orbit  must  be  circular 419 

207.  More  general  types  of  motion  for  the  finite  bodies 420 

208.  Numerical  Example  4 422 

CHAPTER  XIV.  CERTAIN  PERIODIC  ORBITS  OF  k  FINITE  BODIES  REVOLVING 
ABOUT  A  RELATIVELY  LARGE  CENTRAL  MASS.     (By  F.  L.  GRIFFIN.) 

209.  The  problem 425 

210.  The  differential  equations 427 

211.  Symmetry  theorem 427 

212.  Conditions  for  periodic  solutions 428 

213.  Nature  of  the  periodicity  conditions 429 

214.  Integration  of  the  differential  equations  as  power  series  in  parameters. . . .  430 

215.  Existence  of  periodic  orbits  of  Type  1 434 

216.  Method  of  construction  of  solutions.     Type  1 438 

217.  Concerning  orbits  of  Type  II 442 

218.  Concerning  orbits  of  Type  III 447 

219.  Concerning  lacunary  spaces 450 

220.  Jupiter's  satellites  I,  II,  and  III 452 

221.  Orbits  about  an  oblate  central  body 455 

CHAPTER  XV.  CLOSED  ORBITS  OF  EJECTION  AND  RELATED  PERIODIC  ORBITS. 

222.  Introduction 457 

223.  Ejectional  orbits  in  the  two-body  problem 457 

224.  The  integral 461 

225.  Orbits  of  ejection  in  rotating  axes 461 

226.  Ejectional  orbits  in  the  problem  of  three  bodies 462 

227.  Construction  of  the  solutions  of  ejection 463 

228.  Recursion  formulas  for  solutions 465 

229.  The  conditions  for  existence  of  closed  orbits  of  ejection 469 

230.  Proof  of  the  existence  of  closed  orbits  of  ejection 471 

231.  Conditions  at  an  arbitrary  point  for  an  orbit  of  ejection 473 

232.  Closed  orbits  of  ejection  for  large  values  of  n 477 

233.  Periodic  orbits  related  to  closed  orbits  of  ejection 479 

234.  Periodic  orbits  having  many  near  approaches 483 


COM  K\  I'S.  Mil 

(  iiMiKu    XVI.  -StMiiK-i-  .>K  I'KUIODIC  OHHII-   IN    IHI    l.'i -i  iu<  i  KI>   PKOBLEM 

op  THREE  BODIES.  PAGE. 

2:r>.  Statement  of  problem 485 

•j:5t).  Periodic  satellite  ami  planetary  orbits ...  486 

'2'17.  The  non-cxi>icnce  nf  isolated  (icriodic  orbits 487 

•_':;*.  The  persistence  of  double  orbits  with  changing  mav—ratio  of  the  finite 

bo«lie>  4«.M) 

•J.'W.  <  'iisps  on  ]>criodic  orbits 495 

210.  Periodic  orbits  having  l(M>ps  which  arc  related  to  cusps 497 

'-Ml.  The  (MTsistcnce  of  cusps  with  changing  ina»-ratio  of  the  finite  bodies.  ..  .  499 
J  IJ  Some  pro|x-rties  of  the  jM-riodie  oscillating  stitcllitcs  near  the  e<{iiilateral 

triangular  jwiints 501 

LM::.  The  analytic  continuity  of  the  orbits  about  the  equilateral  triangular  points  504 
'-Ml.  The  existence  of  jteriodie  orbits  about  the  equilateral  triangular  points  for 

large  values  of  /i 505 

_M."i.  Numerical  periodic  orbits  about  equilateral  triangular  |M>ints 507 

•Jlti.  ( 'loscd  orbit-,  of  ejection  for  large  values  of  ^  510 

•J  17.  i  >rbits  of  ejection  from  1  n  and  collision  with  n 515 

_Mv  Proof  of  the  existence  of  an  infinite  mimtter  of  closed  orbits  of  ejection  and 

of  orbits  of  ejection  and  collision  when  n  =  )  •> 517 

'-M'.t.  ( )n  the  evolution  of  jx-riodic  orbits  about  equilibrium  points 519 

250.  <  >n  the  evolution  of  direct  periodic  satellite  orbits 520 

_'">1 .  ( >n  the  evolution  of  retrograde  periodic  satollit*  orbits 522 

252.  On  the  evolution  of  periodic  orbits  of  superior  planets 524 


PERIODIC  ORBITS 


BY 

F.  R.  MOULTON 


IN    COLI.ABOHATION    WITH 


DANIEL  BUCHANAN,  THOMAS  BUCK,  FRANK  L.  GRIFFIN, 
WILLIAM  R.  LONGLEY  AND  WILLIAM  D.  MAcMILLAN 


CHAPTER  I. 

CERTAIN  THEOREMS  ON  IMPLICIT  FUNCTIONS  AND 
DIFFERENTIAL  EQUATIONS. 


BY  F.  R.  MOULTON  AND  W.  D.  MACMlLLAN. 


I.    SOLUTION  OF  IMPLICIT  FUNCTIONS. 

I.  Formal  Solution  of  Simultaneous  Equations  when  the  Functional 
Determinant  is  Distinct  from  Zero  at  the  Origin.— -In  applying  the  condi- 
tions for  periodicity  of  the  solutions  of  differential  eq nations  after  the  method 
of  Poincare.*  there  will  he  frequent  occasion  to  consider  the  solution  of 

P,(a,,  .  .  .  ,a.  ;M)=O  (i-1,  ...,»),         (1) 

for  a, ,  .  ...  a.  in  terms  of  /i,  where  the  P,  are  power  series  in  the  a,  and  n, 
vanishing  with  o,  =  0,  M  =  0,  but  not  with  a,  =  0,  M^O,  and  converging  for 
|tt;|<r/>0,  \n\<p>0.  We  are  interested  in  only  those  solutions  which 
vanish  with  M:  that  is,  if  we  regard  a,,  .  .  .  ,  a,,  M  as  coordinates  in 
(H  +  l)-space,  in  those  curves  satisfying  (1)  which  pass  through  the  origin. 
Equations  (1)  can  be  satisfied  formally  by  the  series 

o,  =  S^V,  (2) 

where  the  af  are  functions  of  the  coefficients  of  the  P,  which  are  to  be  deter- 
mined. Upon  substituting  (2)  in  (1),  expanding  and  arranging  as  power 
series  in  M,  it  is  found  that 


rWl>  .....  a<*-")]M* 


(3) 


where  the  f  *'  are  polynomials  in  a*,",  .  .  .  ,  a'/  ". 

Upon  assuming  for  the  moment  that  the  series  (2)  converge  and  satisfy 
(1),  it  follows  that  (3)  are  identities  in  p.  Hence  the  coefficients  of  each 
power  of  M  separately  are  zero. 

•Let  Mtlhodct  NoveeUe*  de  la  Mctaniquc  Ctletie.  vol.  I,  chap.  3.  i 


2  PERIODIC    ORBITS. 

From  the  coefficients  of  the  first  power  of  /x  we  get 


SAP.  ao)  =  _  dPj 
da;      '  dji 

i"j 

Since  by  hypothesis  the  functional  determinant 


(t  = 


(4) 


dPi                  d.Pi 

a       '    '    ' 

da. 

dan 

•      •      •      •      y 

otti                    dan 

is  distinct  from  zero  for  a,=  •  •  •  =an=  M  =  0,  equations  (4)  can  be  solved 
uniquely  for  the  a"'.  If  not  all  of  the  dPt/dn  are  zero,  then  not  all  of  the  a(" 
are  zero;  but  if  all  of  the  dPt/dn  are  zero,  then  all  of  the  a("  are  zero. 

Equating  the  coefficients  of  the  second  power  of  M  in  (3)  to  zero,  we  get 


i  =  \, 


the  right  members  of  which  are  completely  known.  The  determinant  of 
the  coefficients  of  the  af  is  A,  and  the  af  are  therefore  uniquely  determined, 
being  all  zero  or  not  all  zero  according  as  the  Ff  are  all  zero  or  not  all  zero. 
And  it  is  seen  from  (3)  that  the  treatment  of  the  general  term  is  entirely  simi- 
lar and  depends  upon  the  same  determinant  A.  Hence,  under  the  hypothesis 
as  to  A,  a  formal  solution  is  possible,  and  it  is  unique. 

2.  Proof  of  Convergence  of  the  Solutions. — In  order  to  prove  the  con- 
vergence of  the  series  (2) ,  consider  the  solution  of  the  comparison  equations 


(t  = 


.n)       (10 


for  /3i ,  .  .  .  ,  /8»  in  terms  of  M>  where  the  Qt  are  power  series  in  the  /8j  and  n, 
vanishing  for  (8;  =  0,  M  =  0,  and  convergent  for  |  ft,  \  <  r  >  0,  /x  |  <  p'  >  0;  and 
where  also  the  coefficients  of  all  terms  beyond  the  linear  in  each  Q,  are  real, 
positive,  independent  of  i,  and  greater  than  the  moduli  of  the  corresponding 
coefficients*  in  the  expansions  of  any  of  the  Pt .  The  character  of  the  coeffi- 
cients of  the  linear  terms  will  be  specified  when  they  are  used. 
Suppose  the  solutions  of  (!')  have  the  form 


(»•-!,.  ..,*).      (2r) 


*For  proof  of  t.he  possibility  of  satisfying  these  conditions  see  Picard's   Traite  d' Analyse,  edition  of 
1905,  vol.  II,  pp.  255-260. 


IMIM.lt   IT     H    \<    flOXS. 


On  substituting  ij'i  in  (!')  and  arranging  in  powers  of  n,  then-  results  a 
-y-t.-iu  of  equations  similar  to  (3).    The  cf,"  an-  determined  by 


'V 


dfi 


0=1 


(4') 


It  is  necessary  now  to  >peeify  the  properiie-  of  the  linear  terms  of 
the  (/.  It  will  he  supposed  first  that  the  MJ,/dn  are  real  and  positive,  and 
that  dQt/dn=  •  •  •  =dQ*/dn  =  dQ/dfji.  It  follows  that  for  fixed  value-  ol 
the  dQt/dfi,  the  values  of  the  .r  sati>fying  I'i  are  proportional  to  dQ/d». 
The  dQ,/(iJ,  will  now  be  so  determined  that  when  dQ,/dn  is  replaced  by  the 
greatest  \dP,/dn\  the  #',"  determined  by  i4'i  shall  be  equal,  positive,  and 
at  least  as  great  as  the  greatest  ja"'!  for  all  values  of  dP,/dn  such  that 

This  must  be  done  in  such  a  way  that  the  determinant 


dQ. 

a/3, 


dQ. 


shall  be  distinct  from  zero  for  #,  =  ^  =  0.  These  conditions  can  be  satisfied  in 
infinitely  many  ways.  A  simple  way  is  to  choose  dQ,/dfij=  —  1  if  j^i  and 
ay,/a/3,=  -(1  +  r).  Then  the  determinant  of  (4')  is  A'  =  (-l)"c"-'  (c+n), 
wliich  can  vanish  only  if  c  =  0  or  c  =  —n,  and  the  solutions  of  (4')  are 


(5) 


(c  +  n)' 


For  any  n  we  can  give  c  such  a  value  that  the  /3*,1'  shall  be  positive  and  at  least 
as  great  as  the  greatest  |  a"'  |. 

The  &?  are  determined  by  equations  of  the  form 


2  *&  =  ~  °?w>  -----  ^I 

° 


, 


(i 


(6) 


It  follows  from  the  properties  of  the  Q, ,  together  with  the  explicit  structure 
of  the  G?  and  the  values  of  the  #",  that 


G?  ^  |  F ? 


(t  =  l,  .  .  .  ,  n), 


and  therefore  that 


?(») 


a<" 


4  PERIODIC    CEBITS. 

It  is  very  easily  shown  by  induction  that  for  every  value  of  the  index  k 


,...,, 

^  |af>| 


Therefore  if  (2')  converge  for  |  /*  |  g  p',  then  also  (2)  converge  for  |  /*  |  ^  p'. 
All  the  conditions  imposed  upon  the  Qt  can  be  satisfied  by  functions 
of  the  form* 


ft-  -cff,- 


(8) 


where  M  is  a  real  positive  constant.    Adding  these  n  equations  and  solving 
for  ft  +  •  •  •  +  ft,,  it  is  found  that 


(9) 


Since  each  p,  ,  and  therefore  the  sum  18!+  •  •  •  +&,  ,  is  zero  for  M  =  0, 
the  negative  sign  must  be  taken  before  the  radical  in  (9).  The  right 
member  of  (9)  can  be  expanded  as  a  converging  power  series  in  M  if  M  I  is 
taken  so  small  that  M  I  <P  and  |/(M)|  <  |<P(M)|>  where 


conditions  which  can  always  be  satisfied,  whatever  may  be  the  values  of  n,  r, 
p,  c,  and  M  .  Moreover,  the  coefficients  of  all  powers  of  M  in  the  expansion 
of  (9)  are  real  and  positive.  Hence  it  follows  that 

0i+  '    '    '  +/3B=M#(M),  (10) 

where  #(M)  is  a  power  series  in  n  whose  coefficients  are  all  positive.  It 
follows  from  (8)  that  all  the  ft  are  equal.  Hence 


f  -----  -A--  (11) 

For  |ju|  sufficiently  small  the  right  members  of  these  equations  are  converging 
power  series  in  /z;  moreover,  they  identically  satisfy  (8).  It  follows  from 
this  result  and  the  second  set  of  (7)  that  p"  >  0  exists  such  that  the  series  (2) 
converge  for  |  M  I  <  p"- 

*Picard's  Traitt  d'Analyse,  loc.  cit. 


IMPLICIT    FUNCTIONS.  5 

3.  Generalization  to  Many  Parameters.     Suppose  the  equations  to  l>e 
solved  arc 

P,  (a, ,...,  a.  ;  MI  ,...,/*•)  =0          («•-!,...,  n),     (12) 

and  that  the  functional  determinant  with  respect  to  the  a,  is  distinct  from 
/en i  for  a,  =  •  •  •  =O.  =  MI=  •  •  •  =M*  =  0.  Then  the  problem  can  be  reduced 
to  that  discussed  in  §§1  and  2  by  letting  n,  =  c,n.  After  the  solutions  have 
Keen  obtained  the  c,  M  everywhere  can  be  replaced  by  M>,  for  the  c,  M  will  occur 
only  in  integral  powers. 

4.  The  Functional  Determinant  Zero,  but  not  All  of  its  First  Minors 
Zero  at  the  Origin.* — Consider  the  equations 

P,(a,,.  .  .  ,  O,;M)=O  (t  =  i ,„),    (13) 

where  the  P,  have  the  properties  imposed  upon  the  P,  of  §1.  Suppose  that 
the  determinant  of  the  linear  terms  in  the  a,  is  zero  for  a,  =  •  •  •  =  O,,  =  M  =0, 
but  that  not  all  of  its  first  minors  are  zero.  It  may  be  supposed  without 
any  loss  of  generality  that  the  determinant  of  the  terms  remaining  after 
deleting  the  last  row  and  column  of  the  linear  terms  is  distinct  from  zero. 
Hence,  as  a  consequence  of  the  theorems  proved  in  §§1,  2,  3,  the  first  n—l 
equations  can  be  solved  uniquely  for  a, ,  .  .  . ,  a»_i  as  power  series  in  a.  and 
n,  vanishing  for  o»  =  M  =  0. 

Suppose  the  solutions  of  the  first  n—l  equations  for  a, ,  .  .  .  ,  a,_,  are 
substituted  in  the  last  equation.  It  will  become  a  function  of  a,  and  M  which 
may  be  written,  omitting  the  now  useless  subscript  on  the  a, , 

P(a,  M)  =  S    2cwaV  =  0  (*  +  j>0).  (14) 

t-O    1-0 

Since  the  determinant  of  the  linear  terms  of  (13)  is  zero,  this  equation  carries 
no  linear  term  in  a.  Suppose  the  term  of  lowest  degree  in  a  alone  is  cwa*. 
Then,  for  each  value  of  M  whose  modulus  is  sufficiently  small  there  are  k 
values  of  a  satisfying  (14)  and,  moreover,  the  modulus  of  M  can  be  taken  so 
small  that  the  moduli  of  the  solutions  for  a  shall  be  as  small  as  one  pleases. f 
Also,  for  each  set  of  values  of  a  =  a,  and  M  whose  moduli  are  sufficiently  small 
there  is  one  set  of  values  a, ,  .  .  .  ,  cu-i  satisfying  the  first  n—l  equations  of 
(13).  Consequently,  for  each  M  whose  modulus  is  sufficiently  small  there  are 
precisely  k  sets  of  values  of  a(,  .  .  .  ,  a,  satisfying  (13).  A  special  discussion 
is  required  to  determine  the  character  of  these  solutions  and  the  method  of 
finding  them.  These  questions  are  taken  up  in  the  immediately  following 
articles. 


The  more  difficult  case,  in  which  all  the  first  minors  of  the  functional  determinant  vanish,  does  not  arise 
in  this  work.  It  has  only  recently  (in  1911)  been  completely  solved  by  MacMillan,  in  a  paper  which  will 
appear  in  Malhrmatwche  Annaltn. 

tWeientrmos,  Abhandlungtn  out  der  Functionenlehre,  p.  107.  Picard,  TraiU  d'Analyie,  vol.  II,  chap.  9, 
|7,  and  chap.  13.  Hardness  and  Morley,  Trcotue  on  the  Theory  of  Function*,  chap.  4. 


6  PERIODIC    ORBITS. 

5.  Case  where  P(a,  M)  =  a*P,(o.  M)~MX  Ps(tL,  /*).—  Suppose  the  P(a,  M) 
of  (14)  has  the  form 

a*P,(a,  M)   -  MA^2(a,M)  =0, 

where  P!  and  P2  are  power  series  in  a  and  n  which  do  not  vanish  for  a  =  /u  =  0 
and  which  converge  for  |  a  |  <  r  >  0,  M  I  <  P  >  0.  Upon  extracting  the  kth 
root,  this  equation  gives 

a  Ql  (a,  fji)  -  t\  M*  Q2  (a,  ju)  =  0, 

where  Qi  and  Q2  are  power  series  in  a  and  p  which  do  not  vanish  for  a  =  ^  =  0 
and  which  converge  for  a  <  r'  >  0,  M  <  P'  >  0,  and  where  77  is  a  fc'A  root  of 
unity.  If  we  let  ju  =  v",  this  equation  takes  the  form  of  those  treated  in  §§  1 
and  2  and  can  be  solved  uniquely  for  a  in  terms  of  v  for  each  rj.  The  k 
solutions  are  obtained  by  taking  for  r?  the  A-  roots  of  unity. 

6.  A  Second  Simple  Case.  —  Suppose  P  (a,  /x)  has  the  form 

P(o,/i)  =  Sc,a*-V+Q(a,M)  =0,  (15) 


J=0 


where  c0^0  and  Q  contains  no  term  of  degree  less  than  k  +  1  in  a  and  ju. 
It  can  be  supposed  without  loss  of  generality  that  CH  =  1  .  Then 

Sc(a*-y=  P0(a,  M)  =   (o-6lM)(o-6,M)   '   '   '   (a-6t/i).  (16) 

i=0 

Suppose  (a  —  bjfj,)  is  a  simple  factor  of  the  homogeneous  polynomial 
Po  (a,  ;u)>  and  exclude  the  trivial  case  in  which  it  is  also  a  factor  of  Q  (a,  ju). 
Then  dPJda  ^  0  for  a  =  b,  /t.  Now  let 

a  =  6,M  +  j8M.  (17) 

After  this  transformation  both  P0  and  Q  are  divisible  by  M*-  After  //  is 
divided  out,  P0  carries  a  term  in  /3  to  the  first  degree  whose  coefficient  is 
not  zero,  and  Q  carries  no  term  independent  of  ju,  but  has  at  least  one  term 
in  fj.  alone,  for  otherwise  P  (a,  /*)  would  be  divisible  by  (a  —  bjfj.).  Conse- 
quently by  §§  1  and  2  the  equation  in  /3  and  M  can  be  solved  for  /3  as  a  con- 
verging power  series  in  /z,  vanishing  for  ju  =  0.  Therefore  a  can  be  expanded 
as  a  converging  power  series  in  /*,  vanishing  with  /x,  for  each  of  the  simple 
roots  of  the  polynomial  P0  (a,  jj.)  =  0.  If  61  ,  .  .  .  ,  6*  are  all  distinct,  the 
expansions  for  the  k  branches  of  the  function  P(a,  n)  which  pass  through  the 
origin  can  be  found  by  this  process.  The  actual  determination  of  the  coeffi- 
cients is  by  the  method  of  §  1  in  the  simple  case  n  =  1  . 

7.  General  Case  of  Power  Series  in  two  Variables.*  —  The  method  of 
treatment  consists  in  reducing  the  equation,  by  suitable  transformations, 
to  forms  of  a  standard  type  from  which  the  solutions  can  be  found.  In  cer- 
tain special  cases  successive  transformations  are  required.  The  analysis  in 

*This  problem  has  been  treated  by  Puiseux,  Nother,  etc.     For  references  and  discussion  see  Harkness 
and  Morley's  Treatise  on  the  Theory  of  Functions,  chap.  4,  and  Crystal's  Algebra,  part  2,  chap.  30. 


IMPLICIT   FUNCTIONS. 


general,  as  well  as  tin-  particular  transformation  required  in  any  >pecial  ca>e 
to  reduce  the  equation  to  the  standard  forms,  is  indicated  must  simply  by 
\t-\vt  oil's  Parallelogram. 

In  constructing  Newton's  parallelogram  it  is  sufficient  to  consider  only 
those  terms  cua'n'  of  P(a,  /it)  for  which  /  ^  k,  j  ^  X,  r^a'  being  the  term  of 
lowest  degree  in  a  alone  and  c«,x  Mv  the  term  of  lowest  degree  in  n  alone. 
Take  a  set  of  rectangular  axes  and  for  each  of  the  trim-  >•„  aV>  0,7*0,  lay 
down  a  degrti  point  whose  coordinates  are  i  and  j.  Then  the  line  passing 
through  t  lie  origin  and  the  point  A-.O)  is  rotated  around  (k.O)  as  a  pole  so  that 
it  moves  along  the  /-axis  in  the  positive  direction  until  it  strikes  at  least  one 
other  degree  point  (it  may,  of  course,  strike  several  simultaneously).  Ixit  the 
one  of  those  which  it  first  strikes  having  the  greatest  j  he  I/,,./,).  Then  the 
line  is  rotated  around  (ii,ji)  in  the  same  direction  until  it  strikes  at  least  one 
other  degree  point.  Letting  the  one  of  these  having  the  greatest  j  be  (i3,j»), 
the  line  is  rotated  around  (i'2)  jj)  until  another  is  encountered.  This  process  is 
continued  until  the  point  (0,  X)  is  reached.  The  number  of  steps  in  the  process 
evidently  can  not  be  greater  than  /,-  or  X.  The  part  of  Newton's  parallelo- 
gram needed  in  discussing  the  character  of  the  function  near  the  origin  is  made 
up  of  the  segments  (k,  0)  to  (t,  j,),  (t,  j,)  to  (it,  jt),  .  .  .  ,  (i,,j,)  to  (0,  X).  For  the 
terms  of  P(a,  n)  corresponding  to  each  one  of  these  segments  there  is,  as  will 
be  shown,  a  transformation  which  throws  P(a,  n)  into  a  standard  form. 

In  order  to  illustrate   Newton's  parallelogram,  consider  the  example 


P(a,  M)=i 

where  Q(a,  M)  contains  only  terms  of  the  seventh  and  higher  degrees  in  a 
and  M-  The  coordinates  of  the  points  in  Newton's  Parallelogram  are  (5, 0), 
(3,1),  (2,2),  (1,3),  and  (0,6),  and  it 
consists  of  three  segments  which  are 
shown  in  Fig.  1. 

Consider  the  segment  (t, ,  jt)  to 
(i»,j»)  and  make  the  transformation          x 

\(3jlj_ 


(5.0) 


where  m  and  n  are  relatively  prime  inte- 
gers. The  terms  ct,ha' //  and  chh a V*  be- 
come respectively  c^h  /3'1  M*  and  c*h  /3'1/**', 
where 

•V-7'1-      (19) 


(2,2) 


Fio.  1. 


If  there  is  another  degree  point  (if,  f)  on  this  segment  its  coordinates 
satisfy  the  equation 


Hence  after  the  transformation  (18)  the  term  cff  a?  p.'  becomes  cvf  /31' 


8  PERIODIC    ORBITS. 

It  follows  from  the  position  of  the  segment  (ii,ji)  to  (i2,  J2)  with  reference 
to  every  other  degree  point,  that  in  the  case  of  any  term  ctj  a  M'  of  P  (a,  M) 
whose  degree  point  is  not  on  this  segment  the  exponents  i  and  j  satisfy  the 
inequality 

i  (J2—ji)  +j  (i\  —  ii)  +  i-iji  —  iji  >  0. 

Consequently,  after  the  transformation  (18)  the  term  c0aV  becomes  c00V ", 
where 


This  discussion  proves  that  after  the  transformation  (18)  the  terms 
belonging  to  the  segment  (ii,ji)  to  (12,32)  contain  y."'  as  a  factor,  and  that  every 
other  term  contains  /z  to  a  higher  power  than  a '.  Since  a'  is  not  necessarily  an 
integer  the  series  will  not  be,  in  general,  a  series  in  integral  powers  of  n,  but 

it  will  be  in  integral  powers  of  //  =  M"-  Hence,  dividing  out  fj.°'  the  series 
becomes 

0  =  chh  0''  +  •  •  •  +  chh  0"  +  M'  P,  (0,  M')-  (20) 

For  ^'  =  0  equation  (20)  becomes 
c,,A  01'  Pis'-''  +  •••+£*!-  cu,0(i  03 -c,)  •   •   •  (0- *,_«,)  =  0.        (21) 

L  Gfc*J 

The  solution  0'!  =  0  is  not  to  be  considered  for  this  transformation  because 
it  belongs  to  the  solutions  obtained  from  the  segments  having  smaller  values 
of  i.  Suppose  j3  —  c,,  is  a  simple  factor  of  (21)  and  let 

0  =  C,  +  T, .  (22) 

Then  the  right  member  of  (20)  becomes  a  power  series  in  7,  and  n',  van- 
ishing with  y,  =  n'  =  0,  and  the  coefficient  of  7,  to  the  first  power  is  distinct 
from  zero.  Therefore,  by  §1,  the  equation  can  be  solved  for  yr  uniquely  as  a 
converging  power  series  in  //,  vanishing  for  //  =  0.  Then,  on  substituting 
back  in  (22)  and  (18),  a  is  expressed  in  integral  powers  of  /*'.  This  is  an 
integral  series  in  M  only  if  a  is  an  integer.  If  c, , ....  c(l_,,  are  all  distinct  we 
obtain  at  this  step  i\  —  i-2  solutions,  and  if  a  is  an  integer  the  number  of  them 
is  precisely  il  —  i2 .  Since  when  a  is  not  an  integer  /*'  has  more  than  one 
determination,  and  since  the  series  obtained  by  the  transformation  (18) 
after  removing  the  factor  n"'  is  not  in  integral  powers  of  n,  it  would  seem 
that  the  number  of  solutions  for  the  segment  is  greater  than  i l  —  ?2 .  But 
it  will  now  be  shown  that  the  number  of  distinct  solutions  is  {,  —  i.2,  whether 
a  is  an  integer  or  not. 

Now  a=  (J2  —  ji)/(ii-i2)=m/n,  where  m  and  n  are  relatively  prime 
integers.  It  is  clear  that  ii-i2  equals  n,  or  is  greater  than  n,  according  asj2-ji 
and  ii  —  i2  do  not  have,  or  have,  a  common  integral  divisor  greater  than  unity. 
Consider  first  the  case  where  ii  —  i2  =  n.  There  can  be  no  degree  point 


IM1M.HTI     Kl   \<    I  IONS. 


i/',/)  on  the  segment  between  (/',,./,)  and  ii-..j-.),  for  its  coordinate-  would 
have  to  satisfy  the  relation  \j.  -j')/(i'  —  i-j)  =  »'//'.  which  is  impossible  when 
i'  —  i1<n.  Therefore  equation  (21)  beeoine- 


who-c  >olution  gives  /,  -  /'..  values  of  J,  differing  only  hy  the  /',  —  1»  roots  of 
unity.  Hence  the  >aine  linal  results  are  obtained  by  using  the  principal  value 
of  n'  in  (18)  and  all  of  these  i,  —  1»  values  of  0  as  are  obtained  if  any  other 
determination  of  //  is  used. 

Suppose  now  i',  —  i-,=  qn.  where  q  is  an  integer.  There  can  not  be  more 
than  q—  1  degree  points  on  the  segment  /,  ,j,)  to  (it,j»),  i.  e.  satisfying  the 
relation  (ji-j')/(i'-i,)=m/n.  Therefore  in  this  case  (21)  is  a  polynomial  in/3* 
of  degree  q,  and  for  each  of  the  solutions  for  0"  there  are  n  values  of  0  obtained 
one  from  the  other  by  multiplying  by  the  nM  roots  of  unity.  Hence  in  this 
case  also  the  final  results  are  the  same  as  are  obtained  by  using  any  other 
of  the  N  determinations  of  n"  '  . 

It  follows  that  in  every  case  all  the  distinct  solutions  are  obtained  by 
taking  the  principal  value  of  n'  ,  and  that  the  number  of  them  for  the 
segment  (;',  ,  j,)  to  (ij  ,  j3)  is  precisely  i,  —  1'2.  In  a  similar  manner  the 
solutions  associated  with  each  of  the  other  segments  can  be  obtained.  The 
whole  number  of  solutions  found  in  this  way  is 

N  =  (fc-t,)  +  (t,  -  1,)  +  •  •  •  +  (t,  -  0)  =  *,  (23) 

which  is  the  number  of  solutions  of  (P  a,  n)=Q  for  a  which  vanish  with  n  =  0. 
In  this  case  the  problem  is  completely  solved. 

Suppose,  however,  that  in  treating  the  terms  belonging  to  the  segment 
(i'i,  ji)  to  (it,j»)  it  is  found  that  c,  =  •  •  •  =  c,  .  The  analysis  above  fails  to 
give  the  solutions  for  these  roots.  In  this  case  the  transformation 

ft  =  c,  +  7  (24) 

is  made,  after  which  the  right  member  of  (20)  is  a  power  series  in  y  and  n'; 
and  for  /*'  =  0,  7'  =  0  is  a  solution.  That  is,  the  equation  is  of  the  same  form 
as  P(a,  M)=O»  only  in  place  of  having  k  zero  roots  for  /*  =  0  there  are  now 
only  p  such  roots.  This  number  p  is  always  less  than  k  except  when  i,  =  k, 
ji  =  Q,  i:,  =  0,  ;'j  =  X,  and  c,  =  Cj=  •  •  •  =c,l_i,.  But  whatever  the  value  of  p 
a  new  Newton's  parallelogram  for  the  -y^'-equation  is  to  be  constructed. 
It  will  depend  upon  terms  of  higher  degree  in  the  original  a/x-equation 
l>ecause  the  terms  which  gave  rise  to  the  p  equal  roots,  c,  ,  .  .  .  ,  c,,  have  been 
concentrated,  so  to  speak,  by  the  transformations  into  the  single  one  •/, 
and  the  parallelogram  depends  upon  the  term  in  M'  alone  of  lowest  degree. 
By  this  step,  or  some  succeeding  one,  the  solutions  will  all  become  distinct 
unless,  indeed,  the  original  P(a,  M)=O  has  two  or  more  solutions  for  a  which 
are  identical  in  /*• 


10  PERIODIC    ORBITS. 

II.    SOLUTIONS  OF  DIFFERENTIAL  EQUATIONS  AS  POWER 
SERIES   IN   PARAMETERS. 

8.  The  Types  of  Equations  Treated. — In  the  course  of  this  work  certain 
types  of  differential  equations  will  arise  and  they  will  be  solved  by  processes 
adapted  to  attaining  their  solutions  in  convenient  forms.    It  will  tend  to 
clearness  and  brevity  of  exposition  of  the  actual  dynamical  problems  to  set 
down  in  advance  those  methods  of  solving  differential  equations  which  will 
be  used,  and  to  state  the  conditions  under  which  the  results  obtained  by 
them  are  valid.     Consequently,  this  section  will  be  devoted  to  these  ques- 
tions without  making  here  any  applications  to  physical  problems. 

The  equations  which  will  be  treated  are  characterized  chiefly  by  being 
analytic  in  the  independent  and  dependent  variables  and  in  certain  parameters 
upon  which  they  depend;  and  the  solutions  are  considered  only  for  those 
values  of  the  variables  and  parameters  for  which  the  equations  are  all  regular. 
In  the  case  where  the  differential  equations  are  linear,  their  coefficients  are 
either  constants  or  periodic  functions  of  the  independent  variable. 

9.  Formal  Solution  of  Differential  Equations  of  Type  I.* — The  dif- 
ferential equations 

dx 

-~-  =  nft  (ar,  ,  .  .  .  ,  xu  ,  M;  0          (i=l  ,  .  .  .  ,  n)      (25) 

will  be  said  to  be  of  the  Type  I  when  the  right  members  have  n  as  a  factor 
and  when  all  the  ft  are  analytic  in  xv  ,  .  .  .,xn,n  and  t,  and  are  regular  at 
the  point  xt  =  at,  n  =  Q ,  for  all  t0^.t^  T.  Then  the  /,  (xl ,  .  .  .  ,  xn,  M;  ') 
can  be  expanded  as  power  series  in  (xt— a,)  and  M  which  will  converge  if 
\xt-ai\<ri>Q  and  |Ai|<P>Ofor  t^t^T. 

Suppose  Xt  =  at  at  t  =  to,  whatever  be  the  value  of  n.    That  is,  suppose 

xt(Q  =  at,  (26) 

in  which  the  letter  under  the  identity  sign  =  indicates  the  parameter  in 
which  the  identity  is  denned. 

Equations  (25)  can  be  solved  formally  as  power  series  in  /*  which  have 

the  form 

xi  =  S  x\*>  n*.  (27) 


J=0 


where  the  x(?  are  functions  of  t.    On  substituting  (27)  in  (25)  and  arranging 
in  powers  of  n,  it  is  found  that 


(28) 

«v+ 

t_i  _, — .          a/i*j 


*See  Moulton's  Introduction  to  Celestial  Mechanics,  pp.  264-272. 


\i-\-I.INI-.  \K    DIFFERENTIAL    EQUATIONS.  11 

If  tin •-«•  scries  MIC  convergent  the  coefficients  of  corresponding  powers  of  n 
in  the-  right  and  left  members  are  equal.  On  assuming  for  the  moment  that 
they  are  convenient,  the  identity  relations  become 


.." 


;/,  -» 

i,  . 

.  .  ,  n), 

(29) 

./.(* 

(°'»  0;  0                  0'™ii  . 

.  .  ,  • 

(30) 

•    .-*              • 

^  f           af 

Qjt  ^d)  i  ojt 

fii^ 

is, 

r).r,  '        aM  " 

wAJ 

'/•'•     .  y 

9/«  v«+  !  V  V    d*/«    - 

V«     3'/ 

_(l)     I      1     &  ft              /qO\ 

'"       ...  ' 

fcfc  *     r  2  ^  ^/  a^fcr, 

"  x'  +  £  W 

;  *  +  2  aM!  ' 

Th«  of  equations  can  be  integrated  sequentially.    From  (29)  we  get 

x<?  =  ««,  (33) 

where  the  a1"'  are  constants  of  integration.    The  right  member*  of  (30)  are 
now  known  functions  of  t,  and  their  solutions  can  be  written 

*',"  -  J'/.  («7,  0;  0  dt  +  «<"  =  F!"(0  +  a<",  (34) 

where  F',"(0  >s  the  primitive  of/,  (a^,  0;  0-     Then  (31),  (32),  •     •  give  in 
order,  similarly, 

«*,  *?  =  ^(0  +  a",  (35) 


In  this  manner  the  process  can  be  continued  as  far  as  may  be  desired. 

10.  Determination  of  the  Constants  of  Integration  in  Type  I.  —  At  each 
step  n  additive  constants  of  integration  are  obtained,  and  they  must  be 
determined  in  terms  of  the  initial  values  of  the  x,.  From  (26),  (27),  (33),  (34), 
(35)  .  .  .  ,  it  follows  that 


Therefore 

al,0)  =  a,  ,          a?  =  -  **»«,)       (j-1,  ...  oo).  (36) 

By  these  equations  all  of  the  constants  of  integration  are  uniquely  determined 
in  terms  of  the  constants  of  the  differential  equations  and  of  the  initial  values 

of  the  dependent  variables. 

II.  Proof  of  the  Convergence  of  the  Solutions  of  Type  I.  —  The 
method  of  integrating  differential  equations  as  power  series  in  parameters 
has  been  in  use  in  more  or  less  explicit  form  since  almost  the  beginnings 
of  celestial  mechanics.  For  example,  in  the  year  1772  Euler  published  his 


12  PERIODIC    ORBITS. 

second  Lunar  Theory,  in  which  he  used  a  process  quite  analogous  to  this;* 
and  the  method  of  computing  the  absolute  perturbations  of  the  elements  of 
the  planetary  orbits  is  virtually  that  of  developing  the  solutions  as  power 
series  in  the  masses.  But  the  actual  determination  of  the  conditions  for  the 
validity  of  the  process  was  not  made  until  Cauchy  published  his  celebrated 
memoirs  on  differential  equations  in  1842.f  The  results  of  Cauchy  were 
extended  by  Poincare  in  his  prize  memoir  on  the  Problem  of  Three  Bodies,! 
and  were  proved  again,  following  Cauchy 's  Calcul  des  Limites,  in  LesMethodes 
Nouvelles  de  la  Mecanique  Celeste,  vol.  I,  pp.  58-63.  The  theorem  will  be 
needed  in  this  work  in  the  form  given  by  Poincare,  viz.: 

// the  ft  (xi ,  .  .  .  ,xn,n;  t)  of  equations  (25)  are  analytic \\mxt,.  .  .,xtt,fj., 
and  t,  and  regular  at  Xi  =  af,  /u  =  0,  for  all  t0  <it^T,  then  p  >  0  can  be  taken  so 
small  that  the  series  (27)  mil  converge  for  all  t0^t^T  provided  \n\<p. 

To  prove  this  theorem  consider  a  comparison  set  of  differential  equations 

jS-Mftfo,.  ..,*",*;$          (i-l,...,n),    (250 

where  the  &  are  analytic  in  yl , .  .  . ,  yn ,  M,  t,  and  regular  at  yt=  \at\=bt ,  ju=0 
for  all  fc  ^.t^T;  and  where,  further,  the  coefficients  of  all  powers  of  y(  —  bt 
and  M  in  the  expansions  of  all  the  <pt  are  real,  positive,  and  greater  than  the 
moduli  of  the  corresponding  coefficients  in  the  expansions  of  the  /«  for  all  the 
values  of  t  under  consideration.  Then  positive  constants  M,  p,  r,,  .  .  .  ,  rn 
exist  such  that  equations  (25')  can  be  written  in  the  form  § 


The  conditions  are  evidently  satisfied  also  by 
~dt  = 


where  r  is  the  smallest  of  rl ,  .  .  .,rn. 

Suppose  now  the  solutions  of  equations  (25'0  are  developed  as  power 
series  in  ^  of  the  form  «, 

11   —  T  ii(n  n1  C?7'\ 

Vi  —  Li  i/<  M  •  \"'  ) 

There  will  be  quadratures  corresponding  to  (29),  (30),  ....      Moreover, 
by  virtue  of  the  hypotheses  on  the  <pt, 


dt 


dt 


"Tisserand's  Mecanique  Celeste,  vol.  Ill,  chapter  6. 

t  See  Cauchy's  Collected  Works,  1st  series,  vol.  VII. 

\Acia  Mathematica,  vol.  XIII,  pp.  5-266. 

||The  assumption  that  the  /,  are  analytic  in  t  is  not  necessary  for  the  demonstration. 

§Picard's  Traiie  d' Analyse,  edition  of  1905,  vol.  II,  pp.  255-260. 


NON-LINEAR    im  KKKKNTIAL   EQUATIONS.  13 

for  alU  ^  t  ^  T.    Therefore  it  follows  that  t/,"  f  \x(?  \  for  all  t*  ^  /  £  T.    Then 
it  is  seen  from  the  form  of  (31)  that 

o> 

j-5 


dt 


dt 


for  /„  ^/  g  7'.  From  this  it  follows  similarly  that  yT^\af?\.  This  process 
can  be  continued  indefinitely,  giving  by  induction  for  the  general  term 

it?  5  I  af?  I  (37) 

for  Iv^t^T.  Consequently.  if  the  right  members  of  (27')  arc  convergent 
.-cries  when  |/x|<p>0  for  J0^/;S7',  then  likewise  arc  the  right  members 
of  (27)  convergent  when  |M|  <p>0  for  the  same  range  in  t. 

It  is  a  simple  matter  to  find  the  explicit  expression  for  (27')  by  a  direct 
integration  of  (25").    Since  t  he  right  members  are  the  same  for  all  i,  we  have 

yi  —  cl  =  yt  —  ct  =  •  •  •  =  y.-c,, 

where  Ci  ,  .  .  .  ,  c,  are  constants  of  integration.  By  the  initial  conditions  it 
follows  that  c,  —  c,  =  6,  —  b,  and  yl  —  b,  =  y,  —  bi.  Let  this  common  value  of 
3/1  —  61  be  y—  b.  Then  each  equation  of  (25")  becomes 

d|/ 
dt 

On  integrating  this  equation  and  determining  the  constant  of  integration  by 
the  condition  that  (y  —  6)=0  at  t  =  t^,  it  is  found  that 


_      _ 
(l  -  H)  (l  -  nfr~6))  ' 


The  solution  of  this  equation  for  (y  —  b)  is 

r  ,  r 


*-/o).  ax  > 

Since  (y  —  6)  =  0  at  t  =  t<>,  the  negative  sign  must  be  taken  before  the  radical. 
It  follows  directly  from  equation (38) that  whatever  finite  values  n,  M,r,p, 
and  T— <o  may  have,  (y—b)  can  be  expanded  as  a  convergent  series  in  M  for 
!  ^  T  provided  the  condition 

2n  M  \u\. 


•0-5) 


p> 
is  satisfied.    This  condition  imposes  the  explicit  limitation 

1 

-  2nM(T-tJ      1  (39) 


14  PERIODIC    ORBITS. 

upon  n,  which  can  always  be  satisfied  by  ^|  <  Mo>  0  for  r>0,  p>0,  and  for 
M  and  T  —  10  finite.  If  these  conditions  are  satisfied,  the  resulting  expression 
for  (y  —  b)  substituted  in  (25"')  leads  to  convergent  series.  Moreover,  the 
series  for  (y  —  6)  satisfies  (25")  and  is  identical  with  (27')  since  (27')  is 
unique.  Consequently  (27')  converges,  and  therefore  also  (27)  if  M  <  Mo  , 
where  //o  is  the  limiting  value  of  M  satisfying  the  inequality  (39)  ,  for  all  t  in 
the  range  t0^t^  T.  The  theorem  is  thus  established. 

12.  Generalization  to  Many  Parameters.  —  The  differential  equations 
may  involve  many  parameters,  /*1?  .  .  .  ,  pk,  instead  of  a  single  parameter  n. 
The  /,  are  supposed  to  be  regular  for  ^  =  ^  =  •  •  •  =  ju*  =  0  for  t0^t^T. 
The  discussion  can  be  thrown  upon  the  preceding  case  by  letting 


After  the  solutions  have  been  found  /3,  M  can  be  everywhere  replaced  by  //,  . 
This  groups  the  terms  of  the  same  degree  in  /i,,  .  .  .  ,  fj.t  together. 

The  equations  can  also  be  integrated  as  multiple  series  in  the  parameters 
Hi,  .  .  .  ,  p.t  without  the  use  of  this  artifice,  and  then  the  constants  of  inte- 
gration can  be  determined  and  the  convergence  proved.  But  the  method  is 
not  essentially  distinct  from  the  other,  and  the  details  may  be  omitted. 

13.  Generalization  of  the  Parameter.—  Suppose  the  differential  equations 
depend  upon  a  single  parameter  M-  It  may  happen  that  this  parameter 
enters  in  two  distinct  ways.  For  example,  it  may  enter  in  one  way  so  that, 
so  far  as  this  way  alone  is  concerned,  the  ft  can  be  expanded  very  simply  as 
power  series  in  ju.  It  may  enter  in  another  way  so  that,  so  far  as  this  way 
alone  is  concerned,  the  expansions  of  the  ft  as  power  series  in  n  are  very 
complex,  or  even  impossible  without  throwing  the  equations  into  an  unde- 
sirable form. 

Under  the  circumstances  thus  described  it  is  sometimes  of  the  highest 
importance  to  generalize  the  parameter.  Where  it  enters  in  the  first  way  it 
is  left  simply  as  the  parameter  /*•  Where  it  enters  in  the  second  way  it 
is  replaced  by  m  to  preserve  the  distinction.  In  forming  the  solutions  H 
is  regarded  as  a  variable  parameter  in  terms  of  which  identity  arguments  are 
made,  while  m  is  regarded  simply  as  a  fixed  number.  The  solutions  obtained 
are  valid  mathematically  for  any  value  of  /j,  whose  modulus  is  sufficiently 
small,  but  they  belong  to  the  original  (physical)  problem  for  only  one 
particular  value  of  M,  viz.,  for  H  =  m.  But  it  will  be  observed  that  when  the 
differential  equations  are  regular  for  a  continuous  range  of  values  of  m  this 
restriction  is  of  no  importance,  provided  the  solutions  converge  for  fi  =  m,  if 
the  literal  value  of  m  has  been  retained  in  the  solutions.* 


*For  a  practical  application  of  this  artifice  see  Moulton's  Introduction  to  Celestial  Mechanics,  pp.  264-5. 


NON-LINEAR    DIKr  KKKNTIAL   EQUATIONS.  15 

14.  Formal  Solution  of  Differential  Equations  of  Type  II. — The  dif- 
ferent i;tl  equations 

^  =  0,  (x, ,  .  .  .  ,  z. ;  0  +  M/.  (*. ,  •  .  .  ,  x, ,  M;  <)  (,'-i „),     (40) 

will  he  said  to  be  of  Ty|>e  II  if 

(a)  the  g,(xt,  .  .  . ,  x.;<)  are  independent  of  n  and  not  identically 

/•TO; 
(6)  the  g,(xt,       .,  x.;  <)  and  /,(*,,  .  .  .,  *.,  M;  0  are  analytic* 

in  X|,  .  .  .,  x»,  n,  and  I; 
(c)  the  0,(:r, ,  .  .  . ,  xm ;  /)  and  f,(xt ,  .  .  .  x. ,  » ;  /)  are  regular  at 

i,  =  x^(t),  M=0,  for  1*3. t£T,  where  the  x'?  are  the  solutions 

of  equations  (40)  for  n  =  0,  and  a;!01  =  at  at  I  =  t0 . 

It  follows  from  these  conditions  that  the  g,  and  /,  can  be  expanded  as  power 
scries  in  (x,-xf)  and  /i,  which  converge  if  \xt— x™\  <r(>0  and  |M|  <  p  >0  for 
all  I  in  the  range  t^t^T. 

Equations  (40)  can  be  solved  formally  as  power  series  in  /*  having  the 
form  „„ 

x.  =  Sx(,V,  (41) 


1-0 


where  the  x(f  are  functions  of  t,  and  where 

*i(Wfo,.  (42) 

Upon  substituting  (41)  in  (40)  and  equating  coefficients  of  correspond- 
ing powers  of  n,  it  is  found  that 


(43) 

(44) 

<j)_  ,     ,„  <„_,  - 

"  * 


,„  <„_,          /,  ,„,  a/, 


-  2  - 


where  the  &?  are  linear  in  the  coefficients  of  the  expansions  of  gt  and  /,  and 
polynomials  in  xf,  .  .  .,z"~".  In  all  the  partial  derivatives  the  x,  arc  replaced 
by*?. 

The  solutions  of  equations  (43)  are 

,  .  .  .  ,  c.  ;  0,  (47) 


The  assumption  that  the  y,  and  f,  are  analytic  in  /  is  not  necessary,  but  is  made  for  simplicity 
becauae  the  condition  is  always  fulfilled  in  the  applioatiomi  which  follow. 


16  PERIODIC    ORBITS. 

where  d  ,  .  .  .  ,  cn  are  the  constants  of  integration  which  can  be  determined 
in  terms  of  the  at  .  Substituting  these  z'°'  in  (44)  and  integrating,  we  obtain 

x\l>  =  SA<'V,;  «)  +  F<»  (0,  (48) 

where  the  A™  are  the  constants  of  integration.  After  these  solutions  are 
found,  equations  (45)  can  be  integrated,  and  this  process  can  be  continued 
to  the  kth  step,  which  gives 


The  (f>fj  (t)  belonging  to  the  complementary  function  are  the  same  for  each 
step,  but  the  Ff  (t),  which  depend  upon  the  right  members  of  the  differential 
equation,  are  in  general  all  different.  The  problem  of  finding  the  FJ0>,  the  ipw 
and  the  Ff  depends  upon  the  explicit  form  of  the  differential  equations,  and 
can  not  be  given  a  general  treatment. 

15.  Determination  of  the  Constants  of  Integration  in  Type  II.  —  At 
each  step  there  are  n  constants  of  integration  introduced  which  can  be 
determined  in  terms  of  the  initial  values  of  the  xt  .  It  follows  from  equa- 
tions (41),  (42),  (47),  (48),  .  .  .  ,  that 


ri(0)  /_  .,  .  t  \    I     ^'      v   A  '*)       ^  /  ~\  _j_  p'*'  /"/  ^  I   * /i  ^fi\ 

r  4    ^Ci  ,  .    .    .  ,  CB  >  "oy  T"    ^^   I  **  •"i    "w  v  °  /    i        «    ^/  I  /*  v    '  •  \""J 

Hence 

•  *.„/'  '  '^"_^(t)      '  ^     '  "  ""'  (51) 

Suppose  the  constants  Ci , .  .  . ,  cn  are  uniquely  determined  in  terms  of  a, 
by  the  first  set  of  equations  of  (51) .  Then  the  F"'  become  completely  defined, 
and  from  the  second  set  of  (51)  the  A(f  are  uniquely  determined  since  the 
determinant  A  =  |  <pv  (to)  \  is  the  determinant  of  a  fundamental  set  of  solutions 
at  a  regular  point  of  the  differential  equations  and  is  therefore  not  zero  (§  18). 
Then  the  Ff  become  entirely  known  and  the  Af  are  determined  by  a  similar 
set  of  linear  equations  whose  determinant  is  the  same  A.  The  whole  process 
is  unique  and  can  be  continued  indefinitely. 

16.  Proof  of  the  Convergence  of  the  Solutions  of  Type  II. — Consider 
the  comparison  set  of  differential  equations 

-jf  =  &  (yi ,  •  •  • ,  y*',  t)  +  M  tt  (yi  >  •  •  •  >y<»  M;  f),          (40') 

where  the  conditions  corresponding  to  (a),  (6),  (c),  and  (42)  of  §14  are 
satisfied,  and  where,  in  addition,  the  coefficients  of  the  expansions  of  the  <pt 
and  the  ^  as  power  series  in  (yt  —  yf]  and  M  are  real,  positive,  and  greater 
than  the  moduli  of  the  corresponding  coefficients  in  the  expansions  of  the  gt 
and  the  ft  for  all  t  in  the  interval  t0  ^  t  ^  T.  Suppose  yt  =  \  at  \  =  &,  at  t  =  t0 . 


NON-LINEAR    DIFFERENTIAL   EQUATIONS. 

Kquations  (40' i  will  be  solved  in  the  form 


17 


(42') 


where  the  //;'  are  function-  of  /  to  he   determined.       It  will  he  shown  that 
tlie  //,   are  real  and  positive,  and  that 

,^t^T.  (52) 


The  i(,0)  and  y(°'  are  defined  by 


(53) 


Since,  by  hypothesis,  the  integrands  of  the  second  set  of  equations  are  real, 
positive,  and  greater  than  the  maximum  values  of  the  moduli  of  the  inte- 
grands in  the  first  set  of  equations  in  the  interval  to  ^  t  £  T,  it  follows  that 
in  the  whole  interval  yf  >  \a?\. 
The  z'"  and  y("  are  defined  1  >y 


dt 


dt 


a  ) 


, 


(54) 


It  follows  from  (48)  and  (51)  that  x™  =  0  at  t  =  /„ .  Similarly  y',"  =  0 
at  t  =  I* .  Equations  (54)  can  be  solved  by  Picard's  approximation  process.* 
Let  x(£  and  y'i'  be  the  k"  approximations  to  x™  and  y'"-  Then 


(t-1,  .  .   .  ,  w), 


(55) 


(5 


•  •  •  '  "•''  0; 


(56) 


'  •"•'•'•  y"'  0; 


(57) 


•TraiU  tTAnalyte,  vol.  II,  edition  of  1906,  p.  340. 


18  PERIODIC    ORBITS. 

It  follows  from  (55)  and  the  relations  between  the  /,  and  the  ^  that 
y(tl  >  |z"il  for  t0  <!  i  ^  T.  Then,  making  use  of  the  relations  between  the 
coefficients  of  the  expansions  of  the  g,  and  the  <pt  ,  it  follows  from  (56)  that 
y™  ^  z<y|  for  ta  5!  t  ^  T;  and  from  the  method  of  forming  the  successive 
approximations  it  is  seen  that,  y"l  >  \  x™k  for  t0  ^  t  ^  T,  for  all  k. 

Now  Picard  has  shown*  that  Urn  x%  =  z("  for  a  sufficiently  restricted 
range  of  values  of  t.  But  equations  (44)  being  linear,  the  range  of  values 
for  t  is  precisely  that  for  which  the  differential  equations  are  valid.  t  There- 
fore we  conclude  that  y?  >  \  x™  for  t0^.t^T.  The  corresponding  relation 
between  y™  and  x(f  can  be  proved  in  the  same  manner,  and  the  process  can 
be  continued  step  by  step  indefinitely.  Consequently  the  inequalities  (52) 
are  established.  Hence,  if  the  series  (42')  converge  when  |/*|  <  p',  then  the 
series  (42)  also  converge  when  |  n  \  <  p'  f  or  t0  <:  t  ^  T. 

Since 


it  follows  from  the  reference  given  in  §  1  1  that  the  conditions  imposed  upon 
(40')  can  be  satisfied  by  equations  of  the  formt 


As  a  consequence  of  these  equations  (yf  —  y(f)  =  (ys  —  yf)  +  c, ,  where  the 
Cj  are  constants.    Since  yt  =  y^  at  t  =  t0 ,  it  follows  that  c}  =  0.    Now  let 

(59) 


Then,  upon  taking  the  sum  of  equations  (58)  with  respect  to  i,  we  get 

dz 


(60) 


On  integrating  this  equation  and  determining  the  constant  of  integration  by 
the  condition  that  z  =  p.  at  t  =  t0  ,  it  is  found  that 

Mn(t-t0) 


*Loc.  tit.  \Traite  d' Analyse,  vol.  Ill,  edition  of  1894,  p.  91. 

JSee  Les  Methodes  Nouvelles  de  la  Mecanique  Celeste,  vol.  I,  p.  60. 


NON-LINEAK    IHKKKKKM  I At.    KQUATION8.  19 

Solving  this  equation  a  no!  determining  the  sign  of  the  radical  so  that  z  =  n  at 
t  =  I,,,  the  expression  for  r  heroine- 


(62) 

where 

„  Mn 

A  = 


It  follows  from  (62)  that  if  |M!  <  P,  |M|  <  1,  and   T^r^i        <  1.  then  z 

can  be  expanded  as  a  converging  power  series  in  M  for  t^^t^T,  and  that  in 
this  range  for  /  the  values  of  z  are  such  that  the  expansion  of  the  right  mem- 
ber of  (60)  as  a  power  series  in  z  also  converges.  ( 'onsequently,  the  y,  and  x, 
satisfying  (40')  and  (40)  respectively  can  also  be  expanded  as  converging 
series  in  M  for  all  t  in  the  interval  10^.  t  ^  T. 

The  point  to  be  noted  in  these  results  is  that  when  the  differential  equa- 
tions are  of  the  Types  I  or  II,  as  defined  above,and  when  the  interval  T— 1<>  has 
Keen  chosen  in  advance  and  kept  fixed,  then  the  parameter  M.  in  which  the 
solutions  are  developed,  can  be  taken  so  small  in  absolute  value  that  the 
series  in  which  the  solutions  are  expressed  will  all  converge  in  the  whole 
interval  £>  ^  t  £  T. 

As  in  equations  of  Type  I ,  there  may  be  many  parameters,  MI  ,  M* •  •  •  •  >M* » 
instead  of  a  single  one.  The  treatment  can  be  reduced  to  the  case  of  the  single 
one,  just  as  in  the  preceding  case. 

The  parameter  can  be  generalized  precisely  as  was  explained  in  §13.  It 
is  obvious  that  if  there  are  many  parameters  they  may  all  be  generalized. 
Since  the  generalization  can  be  made  in  an  infinite  number  of  ways,  a  great 
variety  of  possible  expansions  for  these  solutions  is  secured. 

17.  Case  of  Homogeneous  Linear  Equations.  While  the  linear  equa- 
tions are  included  in  those  already  treated,  they  deserve  some  special  attention 
for  the  reason  that  in  their  solutions  the  values  of  M  are  not  restricted  by  so 
many  conditions.  Consider  the  equations 


w 

ill-      i-i 

where  the  0,,  are  expansible  as  power  series  in  n  of  the  form 

etl=  ztfV, 

t-o 


20  PERIODIC    ORBITS. 

which  converge  if  | M|  <  P  for  t0^.  t  ^  T .    Suppose  o;,  =  oj  at  t  =  /0.    Then  the 
solutions  can  be  developed  as  power  series  of  the  form 

xt  =  2  x?  /,  (64) 

t  =  0 

precisely  as  in  §  14. 

To  find  the  realm  of  convergence  in  /u  of  (64) ,  consider  a  comparison  set  of 

differential  equations 

dn        " 

-g!  =  2  *„(«)!/,,  (63') 

Ult  ,/=! 

where  the  \f/tj  are  expansible  as  power  series  in  n  of  the  form 


which  converge  provided  |  /j,  \  <  p  for  t0  <  t^  T.    Suppose  also  that  ^(*  >  \6*}\  for 
t0  ^  t^  T.    Develop  the  solutions  of  (630  m  the  form 

00 

jfr-^itfV.  (640 

It  can  be  shown  by  the  method  used  in  proving  the  inequalities  given 
in  (52)  that  if  yfa)  =  |a,|  =  bt,  then  yf  >  x<?\  for  t»  ^  t£  T. 

The  conditions  imposed  on  (630  are  satisfied  by  the  equations 

*/._  y    ¥ 

dt  "  2/  l_v.y»  (63  ) 

p 

in  which  M  is  the  maximum  value  of  the  \Qfi  for  to  ^  t  ^  T.    It  follows  that 
(yi—bt)  =  (yj—bj).    Let  the  common  value  be  (y  —  b).     Then  (63'0  becomes 


p  p 

The  solution  of  this  equation  satisfying  the  initial  conditions  is 


(64-0 

P 

Hence  ?/,  and  therefore  yt  and  x<  ,  can  be  expanded  as  a  power  series  in  ju  con- 
verging for  |  M  |  <  p  f  or  to  ^  t^  T.  That  is,  when  the  differential  equations  are 
linear  the  realm  of  convergence  of  the  solutions  in  the  parameter  n  is  precisely  the 
same  as  that  of  the  coefficients  of  the  differential  equations.  Therefore,  in  those 
simple  cases  in  which  the  original  equations  are  polynomials  in  M,  the  solu- 
tions converge  for  all  finite  values  of  /*.* 

*See  Memoire  sur  les  Groupes  des  Equations  Linfaires,  by  PoincarS,  Ada  Mathemalica,  vol.  IV,  p.  212. 


LINK  Kit    DIFFERENTIAL    EQUA  I  l<  >N  -. 


21 


III.    HOMOGENEOUS  LINEAR  DIFFERENTIAL  EQUATIONS  WITH 

PERIODIC  COEFFICIENTS. 

18.  The  Determinant  of  a  Fundamental  Set  of  Solutions. — Suppose 

where  x[  is  the  derivative  of  x,  with  respect  to  t,  is  the  set  of  linear  homo- 
geneous differential  equations  under  consideration,  and  let 

.  /j\  / j\  _..      *      ft\  I  It 

be  a  fundamental  set  of  its  solutions.  The  determinant  of  this  set  of  solutions 
may  be  denoted  by 

A-|*,|.  (67) 

It  will  be  shown  that  A  can  not  vanish  for  any  t  for  which  the  0,,  are  all 
regular.  In  the  applications  which  follow,  the  0V  are  analytic  in  t  and  in 
general  regular  for  all  finite  values  of  t. 

The  result  of  differentiating  A  with  respect  to  t  is 


A' 


where  the  index  k  denotes  that  in  the  k*  column  the  <pti  are  replaced  by  the 
derivatives  of  the  ^tt  with  respect  to  t.     But  it  follows  from  (65)  that 


Hence  A'  can  be  written 


^ 


» <f>u  > 

»  <Pa  , 

>  <f»  , 


<(>*• 


68) 


,  V- 


The  n  determinants  (68)  can  be  expanded  according  to  the  elements  0,,. 

The  result  is 

»  *       » 

(_!/-!         t-1 


22  PERIODIC    ORBITS. 

where  Au  is  the  minor  of  the  element  <pik  in  A.    But  it  is  known  from  the 

n 

theory  of  determinants  that  S  (  —  l)l+i  <plk  A«  is  zero  when  j  j±  i,  and  equal  to  A 

tml 

when  j  =  i.    Therefore* 

A'  =  A  S  0fl  , 
i=i 

whence 


where  A0  is  the  value  of  A  at  t  =  0.  The  initial  conditions  are  taken  so  that 
AOT^O.  Thus  A  can  vanish  or  become  infinite  only  at  the  singularities  of  the 
coefficients  of  the  main  diagonal  of  the  differential  equations. 

n 

If  2  0«  =  0  the  exponent  vanishes  and  the  determinant  reduces  to  the 
<=i 

constant  A0  .  If  the  differential  equations  were  originally  of  the  second  order, 
having  the  form  usually  arising  in  celestial  mechanics 


x*=  2 


they  are  equivalent  to  the  system 


which  has  the  form  of  equations  (65)  .    Since  every  0«  of  this  set  of  equations 
is  zero,  the  determinant  of  any  fundamental  set  of  their  solutions  is  a  constant. 

19.  The  Character  of  the  Solutions  of  a  Set  of  Linear  Homogeneous 
Differential  Equations  with  Uniform  Periodic  Coefficients.  —  Linear  dif- 
ferential equations  with  simply  periodic  coefficients  were  first  treated  by 
Hillf  in  one  of  his  celebrated  memoirs  on  the  lunar  theory.  About  the 
same  time  HermiteJ  discovered  the  form  of  the  solution  of  Lame's  equation, 
which  has  a  doubly  periodic  coefficient.  Starting  from  the  results  obtained  by 
Hermite,  Picardj)  showed  that  in  general  a  fundamental  set  of  solutions  of  a 
linear  differential  equation  of  the  n"1  order  with  doubly  periodic  coefficients 
of  the  first  kind  can  be  expressed  by  means  of  doubly  periodic  functions  of  the 
second  kind.  In  1883,  Floquet§  published  a  complete  discussion  of  the  char- 
acter of  the  solutions  of  a  linear  differential  equation  of  the  n"1  order  which  has 
simply  periodic  coefficients.  In  this  memoir  Floquet  gave  not  only  the  form 
of  the  solutions  in  general,  but  he  considered  in  detail  the  forms  of  the  solu- 
tions when  the  fundamental  equation  has  multiple  roots.  The  forms  of  the 
solutions  being  thus  known,  the  efforts  of  later  writers  have  been  directed 

'Equation  (69)  was  first  developed  by  Jacob!  in  a  somewhat  different  connection,  Collected  Works, 
vol.  IV,  p.  403. 

]The  Collected  Works  of  G.  W.  Hill,  vol.  I,  p.  243;  Ada  Malliematica,  vol.  VIII,  pp.  1—36;  also  published 
at  Cambridge,  Mass,  in  1877. 

IComples  Rendus,  1877  et  seq. 

\\Comptes  Rendus,  1879-80;  Journal  fur  Mathematih,  vol.  90  (1881). 

lAnnales  de  I'Scole  Normak  Superieure,  1883-1884. 


LINEAR    DI1  H-:i:l.MI.\l.    Kijf  \Iln\s.  23 

toward  the  discovery  of  pradical  melhod>  Tor  their  actual  cuiisi ruction, 
principally  when  the  differential  equation  lias  the  form 

^jj+  (ao  +  a,cos<  +  a,cos2<+  •  •  -)x  =  0.  (70) 

Different  methods  for  constructing  solutions  of  this  equation  have  been 
proposed  by  Lindomunn.*  Lindstedt,t  Bruns,J  Callandreau.  Stieltjcs,§  and 
Harzer.* 

In  what  follows  there  will  arise  only  equations  with  simply  periodic 
coefficients  having  the  form  „ 

elixJ  0-1 »),    (71) 


where  the  6lt  are  periodic  functions  of  t  with  the  period  '2r.  It  will  be 
assumed  that  the  Ou  are  uniform  analytic  functions  of  t  and  are  regular 
forO^<^2T.  Let 

*u  =  <p,i  (/),  Xa  =  <pa  (t),    .    .    .  ,  X,.  =  <pt.  (0         (f-1,  .   .   .   ,  n), 

be  a  fundamental  set  of  solutions  which   satisfy  the  initial  conditions 
<p,,  (0)  =  0  if  i^j  and  <ptt  (0)  =  1  .      It  is  clear  that  n  solutions  can  be 
constructed  with  these  n  sets  of  initial  conditions,  and  since  their  deter- 
minant is  unity  at  t  =  0,  they  constitute  a  fundamental  set  of  solutions. 
Now  make  the  transformation 

x,  =  ea'yt,  (72) 

where  a  is  an  undetermined  constant.     The  differential  equations  become 

21  *,*,  (73) 


any  solution  of  which  can  be  written  in  the  form 

y,  =  e-"^A,vtl  (t)  0-1  .....  n),     (74) 

where  the  A,  are  suitably  chosen  constants. 

The  question  arises  whether  it  is  possible  to  determine  a  and  the  A,  in 
such  a  manner  that  the  y^  shall  be  periodic  in  i  with  the  period  2*-.  From  the 
form  of  equations  (73)  it  is  clear  that  sufficient  conditions  for  the  periodicity 
of  the  t/«  are  that  y,(2ir)  =  yt(0)  (1  =  1  .....  n).  On  imposing  these  condi- 
tions upon  (74),  there  results 

0  =  24,  [*,  (2»)  -  <?*"  *„  (0)]      (i  =  1,  .  .  .  ,  n).    (75) 
i-\ 

Either  all  the  A,  are  zero  or  the  determinant  must  vanish.    The  former  case 
is  trivial  and  we  therefore  impose  the  condition 

I*,  (2»)  -«*"*,  (0)|=0, 

•tlatkanaliKhe  Annalen,  vol.  XXII,  (1883),  p.  117-123. 

iAttronomMu  Nackrichtrn  No.   2503(1883).    MtmniretdfFAeati^mtedeSl.Pfttrri>oury,\o\.  XXI,  No.4. 

lAttronomudu  Nadirichlen,  No.   2533,  2553  (1884). 

\\Attnmomitche  Xachrichtcn,  No.   2547  (1884). 

iWwiomwete  Kachnchlm,  No.   2602(1884). 

\AttrmomM*  ffachriehtcn,  Nos.  2860  and  2851  (1888). 


24 


PERIODIC    ORBITS. 


where  <pti(Q)  =  0  if  iVj  and  <p«(0)  =  1.    Putting  e°a 
simply  by  <ptj,  this  determinant  becomes 


=  s  and  denoting  <pti  (2ir) 


<f>21 
<f>3\ 


<f>»\ 


<f>i3 


l    <f>33        8 


=  0. 


(76) 


This  is  an  equation  of  the  nth  degree  in  s,  the  constant  term  of  which  can 
not  vanish  since  it  is  the  determinant  of  a  fundamental  set  of  solutions. 
It  is  known  as  the  fundamental  equation  for  the  period  2?r.  Its  roots  can  be 
neither  zero  nor  infinite,  because  the  coefficient  of  s"  is  unity  and  the  term 
independent  of  s  is  A. 

20.  Solutions  when  the  Roots  of  the  Fundamental  Equation  are  all 
Distinct.  —  Suppose  the  roots  s,,  s2,  .  .  .  ,  sn  of  (76)  are  all  distinct.  Then 
at  least  one  of  the  first  minors  of  (76)  is  distinct  from  zero  when  s  is  put 
equal  tost,  and  therefore  the  ratios  of  the  A,-  are  uniquely  determined  by  (75). 
For  each  s*  a  set  of  y^  is  determined  by  (74)  involving  one  arbitrary  constant. 
Since  this  solution  depends  upon  s*  it  will  be  designated  by  ytk,  and  the 
corresponding  A,  by  Ajk. 

Since  s  =  emr,  the  a  is  uniquely  determined  in  terms  of  s  except  for  the 
additive  constant  v  V  —  l,  where  v  is  an  integer.  In  every  case  the  principal 
value  of  a  can  be  taken,  for  its  other  values  simply  remove  periodic  factors 
from  the  yt  .  Consequently,  for  the  n  values  of  s  there  are  n  values  a,  and 
from  equation  (74)  there  are  n  solutions,  one  for  each  k  from  1  to  n, 

yik  =  e-a«'  S  Ajk  <(>„  (t)          (1  =  1,  .  ,  .  ,  n),        (77) 


3=1 


where  the  ratios  of  the  AJk  are  determined  by  (75).     From  (72),  n  solutions 
of  equations  (71)  are  thus  found,  one  for  each  k, 


(t  =  l,  .  .  .  ,  n), 


(78) 


xik  =  e  "  yit 

where  the  yik  are  periodic  in  t  with  the  period  2ir. 

These  solutions  (78)  form  a  fundamental  set,  for,  if  they  did  not,  there 
would  exist  linear  relations  among  the  xlk  of  the  form 


S  Ct  x(k  (t)  =  0 


1=1 


(79) 


where  not  all  the  Ck  =  0.    Increasing  t  by  2-rr,  it  follows  from  the  conditions 
imposed  upon  the  xt  and  yt  that 


Ctxlt 


=  S  Ctst  xit  (t)  =  0, 


LINEAR    DIFFERENTIAL   EQUATIONS. 


25 


and  similarly  that 


Ct  xtt  (t  +  4x)  =  2  C*  si  zu  (<)  e  0, 


(80) 


*-l  t-1 

Since  the  C*  are  not  all  zero  it  follows  that  the  determinant  of  these  equations 
must  vanish;  that  is, 


1  ,1  ..1  , 

Si  ,  Sj  ,  Sj  , 

«'  e*  «* 

01  ,  St  ,  5j  , 


n  v  (•*-•*> -o 


(81) 


Since,  by  hypothesis,  the  s,  are  distinct  this  relation  can  not  be  satisfied  unless 
some  XU  =  Q.  For  the  sake  of  definiteness  take  first  i=l,  and  suppose 
that  x^=Q,  where  &,  is  some  particular  value  of  the  second  subscript.  But 
equation  (79)  becomes  for  t  =  1 

£(?.*»»  (0-0 

and  there  is  corresponding  to  (81)  an  identity  of  the  form 


n 


From  this  it   is  inferred  that  another  xit ,  say  xu, ,  is  identically  zero. 
Repeating  the  process  n  times,  the  final  conclusion  is 


Upon  starting  from  (79)  for  i  =  2,  the  conclusion  is  reached  in  a  similar 
way  that 

On  repeating  the  process,  corresponding  identities  are  obtained  for  all 
values  of  i  from  1  to  n. 

Now  from  the  identities  z,*=0  (i=l ,  .  .  .  ,  n)  and  from  (77)  and  (78), 
it  follows  that 

A.21  (fill  ^T     *     *      '     ^~  *T.nt  <^in   ^    U) 

"  °'  (82) 


m  =  0. 


These  identities  can  not  all  be  satisfied  unless  each  A,t  =  Q,  for,  at  t  =  0, 
<p,t  =  0  if  i  ?*j  and  <pu=  1.  The  same  result  holds  for  each  value  of  k  from 
1  to  n,  but  by  virtue  of  equations  (75)  and  the  hypothesis  that  (76)  has  simple 
roots,  it  follows  that  for  each  st  there  is  a  solution  in  which  not  all  the  A  A  are 


26 


PERIODIC    ORBITS. 


zero.  If  these  solutions  are  taken,  the  identities  (82)  can  not  be  satisfied  and 
consequently  equations  (79)  can  not  be  satisfied.  Therefore  (78)  constitute 
a  fundamental  set  of  solutions. 

21.  Solutions  when  the  Fundamental  Equation  has  Multiple  Roots.— 
Consider  first  the  case  where  the  fundamental  equation  has  only  two  roots 
equal.  The  notation  can  be  chosen  so  that  s2  =  Si.  There  are  two  cases 
according  as  all,  or  not  all,  of  the  first  minors  of  (76)  vanish  when  S  —  SL 
Suppose  first  that  all  the  first  minors  vanish  for  this  value  of  s.  Since  s =Si  is 
only  a  double  root,  not  all  of  the  second  minors  can  vanish .  Hence  two  of  the  A t 
can  be  taken  arbitrarily  and  (75)  can  be  solved  for  the  remaining  (n  — 2)  of 
them.  Then  equations  (74)  and  (72)  give  the  corresponding  xt.  Since  the  <pi} 
are  linearly  distinct  two  linearly  distinct  values  of  the  yt  can  be  obtained  by 
taking  first  one  of  the  arbitrary  At  equal  to  zero,  and  then  the  other  equal 
to  zero.  Therefore  in  this  case  there  are  two  linearly  distinct  solutions  of 
the  form 


— 


where  the  yti  and  yi2  are  periodic  in  t  with  the  period  2ir. 

If,  however,  not  all  the  first  minors  of  (76)  vanish  for  s  =s1;  there  is  but  a 
single  solution  of  this  form  belonging  to  the  root  st  of  the  fundamental  equa- 
tion. Let  xti  =  eaityn  be  this  solution;  it  will  be  shown  that  the  other  one 
belonging  to  this  root  has  the  form 

Xt2  =  ea'(  (yf2+ tyn~)  ({  =  1,  .  .  .  .  n),          (83) 

where  the  yt2  are  periodic  in  t  with  the  period  2ir. 

Before  proceeding  to  the  demonstration  a  lemma  pertaining  to  a  certain 
type  of  transformation  of  a  fundamental  set  of  solutions  will  be  proved. 
Suppose  the  <?„  constitute  a  fundamental  set  of  solutions.  Then  define  new 
functions  \l/ft  by  the  relations 


(84) 
Akk  =  Q,  for 


Yik  —   ^  -n-ikVu  (.»>»••*>   •   •   •   ,'i). 

The  if/a  also  constitute  a  fundamental  set  of  solutions  provided  no  Akt 
the  determinant  of  the  \{/ik  is 

n 

where  \Ajk\  and  \<pi}\  are  the  determinants  of  the  Ajk  and  <pti  respectively. 
The  determinant  |  <ptj  \  is  distinct  from  zero  and 


•"•11  ,   -"-21  }    -"31    >  •  •  • 

U  ,    A.-II  ,    .".32    ,  •  •  • 

0  ,0  ,     ^33    ,  •  •  • 

0  ,0  ,0 


(85) 


LINKAIt    DIKFKHKNTI  \l      IJ.'I    \II()N8.  27 

which  is  distinct  from  zero  unless  some  A,,  is  zero.  A  special  case,  which  will 
be  used  first,  is  that  when-  all  the  elements  except  those  in  the  first  line  and 
in  the  main  diagonal  are  zero. 

Now  return  to  the  point  under  discussion.  By  hypothesis  not  all  the 
lii>t  minors  of  7ii  v:ini>h  for  ,s  =  .s, .  Let  the  notation  be  chosen  so  that  one 
of  those  which  is  distinct  from  zero  is  formed  from  the  elements  of  the  last 
n  —  1  columns.  Then  .1,  must  be  distinct  from  zero  in  order  not  to  get  the 
trivial  case  in  which  all  the  4 ,  are  zero.  Now  in  placeof  <plf  (i,j=l ,  .  .  .  ,  n) 
as  a  fundamental  set  of  solutions  we  can  take,  as  a  consequence  of  the  lemma, 

ea>'y<i,         *„  (i-1 »»;j-2 n).       (86) 

Any  solution  can  be  expressed  in  the  form 

xt  =  Bte^'yn  +  ZBjv.,  (t-1,  .  .  .  ,n). 

/-i 

Now  make  the  transformation    z,2  =  ea>l  (y,t  +  tyn);  whence 
y,,  =  -  tyn  +  B,  yn  e(a'-a->4+  r^B,*,, . 


Since  by  hypothesis  the  xtl  =  ea>tyn  satisfy  (71),  it  is  found  by  substitution 
that,  if  (83)  are  to  constitute  a  solution,  the  y,,  must  satisfy  the  equations 

y(t  +  a,  ytt  =  £  etj  y,*-yn        (t = i n).  (87) 

Since  t  enters  only  in  the  0(,  and  the  ytl,  which  are  periodic  with  the  period 
2ir,  sufficient  conditions  that  the  ytl  shall  be  periodic  with  the  period  2v  are 


yit  (2v)  -  ytt  (0)  =  -  2r  ytt  (0)  +  ^B,  (<ptl  (2ir)  f*  -'-#»„  (0)]  -  0. 
On  transforming  from  the  exponential  to  s,  these  equations  give 

-2x «,  ytl  (0)  +  J.B,  \v<,  (2»)  -  «, *,/]  =  0,  (88) 


where  6,/  =  0  if  jj*i  and  $,,=  !• 

The  condition  that  equations  (88)  shall  be  consistent  is 

Di  =  |yu(0),*»(2») -«,«,,,  .  .  .  ,^i.(2») -*,«,.| -0, 

where  Z),  is  the  determinant  formed  from  their  coefficients.  This  equation 
is  satisfied,  for  if  the  fundamental  equation  is  formed  as  usual  from  the 
fundamental  set  (86),  it  is  found  that  D=(s-st)D1  =  Q.  Since  D  is  inde- 
pendent of  the  fundamental*  set  from  which  it  is  derived,  and  since  Z)  =  0 
has  the  double  root  s  =  s, ,  it  follows  that  Z),  (s,)  =  0.  Therefore  (88)  can  be 
solved  uniquely  for  the  B,  ,  .  .  .  ,  B».  These  equations  determine  the  y(t , 
and  through  them  the  x,2  in  the  form  given  in  (83). 

•Fucha,  Journal  fur  Matkematik,  vol.  LXVI  (1866),  p.  133. 


28  PERIODIC    ORBITS. 

Now  suppose  s  =  Si  is  a  triple  root  of  the  fundamental  equation,  but  that 
it  is  not  a  quadruple  root.  If  all  its  minors  of  the  first  and  second  order 
vanish  for  s  =  Si  ,  three  of  the  A}  can  be  taken  arbitrarily  and  three  linearly 
distinct  solutions  of  the  form 

xn  =  eait  yn  ,  Xt2  =  ea>t  ya  ,  xi3  =  ettl(  ya 

can  be  determined,  where  the  yn,  yf2,  and  yi3  are  periodic  in  t  with  the  period  2ir. 
If  all  of  the  minors  of  the  first  order  of  the  fundamental  determinant 
vanish,  but  not  all  of  those  of  the  second  order,  then  two  of  the  A,  can  be 
taken  arbitrarily,  and  two  linearly  distinct  solutions  of  the  form 


will  be  obtained,  where  the  yn  and  yi2  are  again  periodic. 

In  order  to  obtain  a  third  solution  associated  with  the  root  st  take  as 
a  new  fundamental  set  of  solutions 

eBl<y,i,  ea'(?/J2,  ViJ  (i=i,  .  .  .  ,  n-  j=3,  .  .  .  ,  n), 

so  that  any  solution  can  be  written  in  the  form 

xt  =  B,  ea>'  yn  +  B,  ea''  ya  +  S  Bj  <ptj          (f  =  i,  .  .  .  ,  „). 

J=3 

Now  make  the  transformation 


whence 


>t  S 


In  a  manner  similar  to  that  in  the  case  just  treated  the  periodicity  conditions 
on  the  yi3  lead  to  the  equations 

0=  -27T  *!  yn  (0)  -  27r  s,  ya  (0)  +  S  B,  [<ptj  (2ir)  -  «„  sj. 

J=3 

As  in  the  preceding  case,  it  is  found  that  for  s=Si  the  B3,  .  .  .,  Bn  are 
uniquely  determined  and  that  the  xi3  have  the  form 

Xa  =  ea><  [ya  +  t  (yil  +  yj],  (89) 

where  the  yn,  ya,  and  yi3  are  periodic  in  t  with  the  period  2-n-. 

Suppose  now  that  not  all  of  the  first  minors  of  the  fundamental  determi- 
nant vanish  for  s=Si.  Then  there  will  be  one  solution  xtt  =  e°"'  yn  and  another 
x{2  —  ea''  (yl2  +  t  yn).  It  will  be  shown  that  in  this  case  the  third  solution 
belonging  to  St  is  of  the  form 

xa  =  ett1'  [ya  +  t  ya  +  f  t  yn}.  (90) 


LINEAR   DIFFERENTIAL   EQUATIONS.  29 

Take  as  a  new  fundamental  set  of  solutions 

ea''t/u ,      e°'(  (ya  +  lytl),       <pt)  (,'=1,  ...,»;  j=3,  ...,«).      (90') 

After  defining  the  x,  by 

j,  =  £,  ea'(  t/,,  +  5,  ea-(  (yrt  +f  y,,)  +  X  B,  <pl}        (i- 1,  ...,»), 

make  the  transformation  (90).     Then  the  ex|>n'->ion>  for  ya  arc 
y«=  -  tya  - 


If  the  a-,3  constitute  a  solution  of  the  original  equations  (71),  the  ylt  must 
satisfy  the  equations 

!/,',  + a,  t/,3  =  26tlyK  -  y,t, 

since  the  y(,  satisfy  (73)  and  the  y,t  satisfy  (87).     Hence  sufficient  conditions 
that  the  yn  shall  be  periodic  are  that 

y,,(2r)-y,,(0)  =  0         (i-i ,  „). 

These  conditions  lead  to  the  equations 

"5  B,  [<flt  (2  T)  -  s,  5,J. 


The  terms  y,i  (0)  in  the  second  column  of  the  determinant  of  the  coefficients 
of  these  equations  evidently  may  be  suppressed.  Let  this  determinant  be 
denoted  by  D, .  In  order  that  these  equations  shall  be  consistent  it  is 
necessary  that  D2  =  0.  This  condition  is  satisfied;  for  if  the  fundamental 
equation  be  formed  from  (90'),  it  is  found  that 

D=(s-8>)t  Dt  . 

But  by  hypothesis  D  admits  (s  =  «,)  as  a  triple  root.  Therefore  Dt  («t)  =  0 
and  the  equations  are  consistent.  Since  s  =  «i  is  a  simple  root  of  Dt  ,  not 
all  of  its  first  minors  are  zero.  Therefore  the  Bt ,  .  .  .  ,  Bn  are  uniquely 
determined,  and  the  z,3  have  the  form  (90). 

Suppose  «= s,  is  a  root  of  multiplicity  I.  There  is  then  a  group  of  solu- 
tions, /  in  number,  attached  to  this  root.  In  general  this  group  of  solutions 
will  have  the  following  form 


=  ea''ylt  (t-1,  .  .  .  ,n), 


(91) 


30  PERIODIC    ORBITS. 

If  all  the  minors  of  the  fundamental  equation  D  =  0  up  to  the  order 
k  —  l(k^l),  but  not  all  of  order  k,  vanish  for  S=ST,  then  there  are  k  solutions 
of  the  first  form,  i.  e.  of  the  form 

Xu    =    6    '     t/il  ,  Xi2    =6    '    2/f2  )  •      •      •    J  %ik    =&         2/ifc  • 

If  now  the  fundamental  set 

ea>'ytl,  .  .  .  ,eaityik,        ?,.*+,,  .  .  .  ,  ?fc , 
be  taken  and  the  equation  in  s  formed,  it  is  found  that 

~r~\  /  \  t    T~\  f\ 

I J    —    I  o  ™~  o  i  )  K    ~~~    ^  • 

Since  the  roots  of  the  fundamental  equation  are  not  changed  by  adopting 
the  new  fundamental  set  of  solutions,  Dk  =  0  has  s  =  Si  as  a  root  of  multiplicity 
I— k.  Suppose  all  the  minors  of  Dk  of  order  g  —  1,  but  not  all  of  order  g, 
vanish;  then  there  are  g  solutions  of  the  second  form,  viz., 

atT  "        1  a,<  f  "         1 

If  k+g<l,  by  a  similar  change  of  the  fundamental  set  of  solutions,  it  will 
be  found  that 

D  =  (s  —  s)k+°D.     =  0 

Now  Dk+a  =  Q  admits  s  =  s,  as  a  root  of  multiplicity  I—  (k+g)  and  there  is 
a  certain  number  of  solutions  of  the  third  type  of  (91),  depending  upon 
the  order  of  the  minors  of  DK+g  which  do  not  all  vanish  for  s=s,.  Continuing, 
there  is  obtained  finally  I  linearly  independent  solutions  associated  with  s, , 
and  in  a  similar  way  the  solutions  associated  with  the  other  roots  of  the 
fundamental  equation  can  be  found. 

22.  The  Characteristic  Equation  when  the  Coefficients  of  the  Differ- 
ential Equations  are  Expansible  as  Power  Series  in  a  Parameter  n- — In 
the  preceding  discussions  no  explicit  reference  was  made  to  the  parameters 
upon  which  the  00  may  depend.  It  will  be  assumed  now  that  the  6U  are 
expansible  as  power  series  in  /JL  whose  coefficients  separately  are  periodic  in 
t,  and  that  the  series  converge  for  all  finite  values  of  t  if  n\  <p.  It  will  be 
assumed  further  that  6ij  =  au,  where  the  au  are  constants,  for  /z  =  0.  Under 
these  conditions,  which  are  often  realized  in  practice  and  particularly  in  the 
applications  which  follow,  the  discussion  of  the  character  of  the  solutions 
can  be  made  so  as  to  lead  to  a  convenient  method  for  their  practical  con- 
struction. The  discussion  will  depend  upon  the  principles  of  §  19  and  the 
integration  of  the  equations  as  power  series  in  ju- 

Consider  now  the  equations 

"    r  t-. 


LINEAR    DIKKKKKM!  \l.    KQUATIONS.  31 

where  the  a0  are  constants,  the  0,',  are  periodic  in  t  with  the  period  2ir,  and 

oo 

2  0(,*V  converge  for  all  real,  finite  values  of  /  if  \n\<p.    For  /u  =  0  equations 

(92)  admit  xi0'  =  c,ca'"'  as  a  solution,  where  the  r,  are  constants  whose  ratios 
depend  upon  the  coefficients  of  the  differential  equations,  and  a<0>  is  one  of 
the  roots  of  the  characteristic 


fln-a"",  a,, 
a,,          ,  <;., 


.(0) 


=  0. 


(93) 


This  equation,  which  is  of  the  n"  degree  in  a<0),  has  n  roots,  a™,  .  .  .  ,  a™.  If 
these  roots  are  all  distinct  there  exists  a  fundamental  set  of  solutions  of 
the  form 

xt?  =  ct)  e*>  (»-i, n ;  j  =  l, n).      (94) 

If  two  of  the  roots  are  equal,  say  a{0)  =  aJ0),  a  fundamental  set  of  solutions 
is  obtained  by  taking 


r<*>-r   A      T<O)  _  /     +tP\  e         x    =  c  e'  x    =  c,  e  • 

*ti  ~cn"         t  *i»   ~"  \ca    i    t'ci\)  €          >  *ia         nte         >  •    •    •  )  *•   '     vfc« 

Suppose  the  roots  of  (93)  are  all  distinct  and  that  the  fundamental 
set  of  solutions  is  (94);  then,  for  n  distinct  from  zero,  the  complete  solutions 
of  (92)  are 

x,  =  2  A,  [c(,ea>"'+  S 


(f-i,  .  .  .  ,  n),     (95) 


where,  by  §17,  the  series  Zx*  MU)  converge  for  any  preassigned  finite  range 

for  t  if  |  M  I  <  P  •  Without  loss  of  generality  the  initial  conditions  can  be 
taken  so  that  the  determinant  of  the  c,,  is  unity  and  x™(Q)  =0.  As  before, 
the  transformation 


=  eat  yt 


is  made,  and  the  equations  corresponding  to  (92)  and  (95)  are  respectively 

(96) 


The  conditions  that  the  j/,  shall  be  periodic  with  the  period  2ir,  viz., 
l/,(2T)-j/,(0)  =  0,  give 


0  = 


(97) 


32 


PERIODIC    ORBITS. 


A  = 


M 


Since  the  Aj  must  not  all  be  zero,  the  determinant  of  their  coefficients  must 
vanish,  whence 

*]  |  -  0.  (98) 

This  equation  has  an  infinite  number  of  solutions,  for  if  a  =  a.j  is  a  solution, 
then  also  is  a  =  ai  +  v  V—  1,  v  any  integer.  The  fundamental  equation 
corresponding  to  (76)  is  obtained  by  the  transformation  e2air  =  s.  If  the  values 
of  s  satisfying  the  fundamental  equation  are  distinct,  the  corresponding 
values  of  a  are  distinct  but  not  the  converse,  for  if  two  values  of  a  differ  by  an 
imaginary  integer  the  corresponding  values  of  ,s  are  equal.  Only  those 
values  of  a  will  be  taken  which  reduce  to  the  ac;0)  for  M  =  0,  the  a(f  being 
uniquely  determined  by  (93)  . 

Suppose  now  that  two  of  the  roots  of  the  characteristic  equation,  say 
a?  and  a?,  are  equal.     Then  the  solutions  of  (92)  have  in  general  the  form 


xt  = 


J=3 


(99) 


The  exception  to  this  general  form  is  that  tca  may  be  absent  from  the 
second  term  for  i  =  1 ,  .  .  . ,  n,  and  this  possibility  must  be  considered  at 
those  places  where  it  makes  differences  in  the  discussion. 

After  making  the  transformation  x(=eatyt,  the  solutions  for  the  yt  are 


_ 


t 


.7=3 


(100) 


The  conditions  for  the  periodicity  of  the  yt,  viz.,  yi(2ir')—  yt(0)  =0,  lead  to 
the  determinant 


A  = 


'  S 


=  0, 


(101) 


where  the  elements  which  are  not  written  are  of  the  same  form  as  those  in  (98)  . 
If,  for  M  =  0,  the  characteristic  equation  has  a  root  of  higher  order  of 
multiplicity,  the  fundamental  equation  is  formed  in  a  similar  manner. 

23.  Solutions  when  af,  o 


, 


af  are  Distinct  and  their  Differ- 
ences are  not  Congruent  to  Zero  mod.  v/^T.— The  part  of  (98)  independent 
of  M  is 

(102) 


and  the  determinant  ct}\  is  unity. 


LINKAU    UIKKKKKVriAI,    KlJt'ATIOXS.  33 

If.  in  any  particular  case,  (98)  wen-  an  identity  in  n  its  n  solutions  would 

be  simply  o  =  o(j".     In  <"i.-e  it  is  not  an  identity,  let 


(103) 

and  A  beoomei 


where  f'«(&,/i)  is  a  -erie>  in  /3»  and  /i,  converging  for  |0»|  finite  and  |/ 

Since,  by  hypothec-,  i,i>  i>  an  imaginary  integer,  the  expan- 

sion of  (104)  as  a  power  series  in  /3*  and  yu  contains  a  term  in  ^  of  the  first 
degree  and  no  term  independent  of  both  &  and  //.  Therefore  (see  §§1  and 
2)  it  can  be  solved  uniquely  for  &  as  a  i>ower  series  in  M  of  the  form 


M).  (105) 

Substituting  this  value  of  &  in  (103)  and  the  resulting  a  in  (97),  n  homo- 
geneous linear  relations  among  Al  ,  .  .  .  ,  An  are  obtained  whose  determi- 
nant vanishes,  but  for  n  sufficiently  small  not  all  of  its  first  minors  vanish, 
since  the  roots  of  the  determinant  set  equal  to  zero  were  all  distinct  for  M  =  0  . 
Therefore  the  ratios  of  the  ^4;are  uniquely  determined  as  power  series  in  n, 
converging  for  |/u|  sufficiently  small.  When  the  ratios  of  the  A,  have  been 
determined,  the  yu  are  determined  as  power  series  in  //,  and  the  coefficient  of 
each  power  of  p.  separately  is  periodic  in  t.  A  solution  is  found  similarly 
for  each  of. 

The  origin  of  the  singularities  which  determine  the  radii  of  convergence 
of  the  final  solution  series  is  known.  If  p  is  the  smallest  true  radius  of 
convergence  of  the  original  solutions  (95)  as  t  varies  from  0  to  2*,  then,  in 
general,  the  final  solutions  will  converge  only  if  |  n  \  <  p.  Consider  the  funda- 
mental equation,  A(s,  n)=Q,  which  is  a  polynomial  in  s  of  degree  n  and  a 
power  series  in  n  converging  if  \p\  <p.  From  the  algebraic  character  of  A  it 
follows  that  the  only  singularities  introduced  by  solving  for  s  in  terms  of  n 
are  branch-points,  which  are  determined  by  the  simultaneous  equations 

A(s,M)  =  0,          ^  =  0.  (106) 

The  variable  s  can  be  eliminated  from  these  equations  by  rational  procc 
and  the  eliminant  will  converge  if  |/*|  <p.     Its  zeros  are  branch-points  for  s 
as  defined  by  A(s,  M)  =0- 

The  zeros  of  the  eliminant  which  lie  within  |M|  =P  can  be  found  in  any 
particular  numerical  case  by  Picard's  extension  of  Kronecker's  method*  pro- 
vided the  zeros  are  all  simple.  If  there  is  a  zero  at  /*  =  n*  ,  then  the  solutions 
for  s  as  a  power  series  in  M  converge  only  if  |/*[  <  |/i«|.  If  there  is  no  ^  the 
limit  remains  p. 

•Picard's  TraM  fAnalyte,  vol.  II,  ch»p.  7. 


34  PERIODIC    ORBITS. 

Now  consider  ft  as  a  function  of  /z  through  its  relation  with  s,  viz., 

st  =  e^"  +^"'  *  =  ea>  T  •  e  t?r.  If  s»  has  a  branch-point  for  /j.  =  /*„,  then  ft  also  has 
a  branch-point  at  the  same  place  since  dj3/ds  =  1/2x8  is  distinct  from  zero 
for  all  finite  values  of  s.  Therefore  the  series  for  ft  converges  only  if  \/j,  <  \JJLO\  . 

The  root  st  iss,  =  sf+  2  4V  and  a<0)  +  ft  =  (  1/2*-)  log  [a™  +  2  s™  /*']. 

(=1  i=i 

If  for  any  MI  such  that  |  MI  I  <  P  we  have  |  sf  \  =  \  S  s™  M!  I  ,  then  ft  has  an 

essential  singularity  at  yu  =  /xj  ,  and  the  series  for  it  converges  only  if  |/x  <  /ii  |  . 
The  zeros  determining  these  singularities  can  also  be  found  in  a  special 
numerical  case  by  Picard's  method.  When  \/j.  satisfies  the  inequalities 
imposed  by  these  various  possible  singularities,  the  solutions  are  convergent 
for  all  finite  values  of  t. 

24.  Solutions  when  no  two  af  are  equal  but  when  af  —  a™  is  Congruent 
to  Zero  mod.  V—l.  —  Suppose  two  roots  of  the  characteristic  equation  for 
p.  =  0,  say  dj0>  and  af,  differ  only  by  an  imaginary  integer,  and  that  there 
is  no  other  such  congruence  among  them.  Then  the  equation  corresponding 
to  (104)  becomes 


-  e1-        ')  +ft  MFt(ft  ,  M)  +M2F2(ft  ,  M)  =  0.    (107) 

/=3 

The  term  of  lowest  degree  in  ft  alone  is  4.ir*P\.  The  term  of  lowest  degree 
in  M  alone  is  at  least  of  the  second  degree,  and  will  in  general  be  precisely 
of  the  second  degree.  This  follows  from  the  fact  that  every  term  in  every 
element  of  the  first  two  columns  of  the  determinant  (98)  contains  in  this 
special  case  either  ft  or  /x  as  a  factor.  In  order  to  get  the  terms  in  fj,  alone, 
those  involving  ft  are  suppressed,  and  then  the  conclusion  follows  from  the 
fact  that  every  term  in  the  expansion  of  the  determinant  contains  one 
term  from  each  of  the  first  two  columns.  In  a  similar  way  if  p  of  the  a("} 
are  congruent  to  zero  mod.  V  —  l,  then  the  term  of  lowest  degree  in  ft  alone 
is  exactly  of  degree  p,  and  in  /x  alone  it  is  at  least  of  degree  p. 

Consider  the  expansion  of  (107),  which  may  be  written  in  the  form 


M2+    •    •    '    =0, 

where  yn,  702,  are  constants.     The  quadratic  terms  can  be  factored,  giving 
(ft  —  &!/x)(ft  —  62M)  +  terms  of  higher  degree  =0. 

If  61  and  62  are  distinct,  as  will  in  general  be  the  case,  the  two  solutions 
of  (107)  are  then  (see  §6), 

/9n  =  6iM+M'Pi(M),  ft2  =  &2M  +  M2JP2(M),  (108) 

where  PI  and  P2  are  power  series  in  n.     If  bi  =  b2  the  solutions  are  power 
series  in  VjZ  or  p,,  depending  upon  the  terms  of  higher  degree.      If  702  is 


UNKAK    DlKKKHKVn.VL   EQUATIONS.  35 

zero  at  lea.-t  mic  <if  the  solution.-  starts  with  n  term  of  degree  higher  than 
the  first  in  M.  If  the  first  term  in  n  alone  is  »'  and  if  7,,  is  zero,  then 
the  solution  has  the  form 


But  in  general  the  solutions  arc  of  the  type  (108),  and  no  other  social  cases 
will  be  considered  in  detail;  they  can  all  be  treated  by  the  principles  of 
§§6  and  7.  Thus,  starting  from  the  root  a,"  of  i  ho  characteristic  equation, 
two  solutions  are  obtained.  But  it  follows  from  the  form  of  equations 
(98)  and  (104)  that  if  the  start  were  made  from  the  root  aj",  the  same  values 
for  0,  would  be  found. 

The  other  &(fc  =  3,  .  .  .  ,  w)  are  found  as  in  the  preceding  case,  the 
solutions  from  them  are  formed  in  the  same  way,  and  their  realm  of 
convergence  is  limited  by  possible  singularities  of  the  same  types. 

25.  Solutions  when,  for  n  =  0,  the  Characteristic  Equation  has  a  Mul- 
tiple Root.  Suppose  only  two  roots  are  equal,  say  a"'  =  a™',  and  that 
there  are  none  of  the  congruences  treated  above.  Then,  for  /*  =  0.  equation 
(101)  becomes 


.  0 

+2TC,,,.  .  ., 
which  easily  reduces  to 

n 

since  the  determinant  |c,,|  is  unity. 

After  the  substitution  a  =  a™+tit  is  made  in  (109)  and  the  result 
expanded,  it  is  found  that  the  term  of  lowest  degree  in  0i  alone  is  4*-J/3*  . 
If  the  determinant  A  for  ai0)  =  a[w  is  of  the  special  form  (98),  the  term  of 
lowest  degree  in  n  alone  is  at  least  of  the  second;  but  if  in  this  case  A  is  of 
the  general  form  (101),  the  term  of  lowest  degree  in  n  alone  is  in  general  of 
the  first.  Except  in  the  special  cases  the  solutions  of  (101)  in  the  vicinity 
of  the  double  root  a(,0)  are  of  the  form 


where  /'  is  a  power  series  in  M',  containing  a  term  independent  of  /*. 

When  the  coefficient  of  n  is  zero  in  the  expansion  of  (101),  the  first 
term  in  n  alone  is  of  at  least  the  second  degree,  and  the  problem  is  of  the 
type  treated  in  the  preceding  article. 

If,  for  n  =  0,  p  roots  of  the  characteristic  equation  are  equal,  then  for 
these  roots  the  expansion  of  (101)  starts  with  ft  as  the  term  of  lowest  degree 
0i  alone,  and  except  in  special  cases  the  lowest  degree  of  terms  in  M  alone  is 


36  PERIODIC   ORBITS. 

the  first.    Consequently  in  general  for  af  =af=   •  •  •   =  <>  the  solutions 
of  (101)  are 


where  e  is  any  pth  root  of  unity. 

Another  case  is  that  in  which  A  =  0  has  a  double  root  identically  in  n, 
the  conditions  for  which  are 

A(a,M)=0,          |£(«,M)=0 

for  all  |M|  sufficiently  small.  Suppose  a2  =  a,  .  If,  for  a  =  0.1  ,  all  the  first  minors 
of  A  are  zero,  the  solutions  of  (97)  for  the  ratios  of  the  At  will  carry  two  arbi- 
traries,  and  the  two  solutions  associated  with  a,  will  be  obtained.  If  not  all 
the  first  minors  of  A  vanish  for  a  =  alt  then  in  this  way  only  one  solution  is 
found.  But  it  is  known  from  the  general  theory  of  §21  that  the  second 
solution  has  the  form 


On  substituting  these  expressions  in  the  differential  equations  and  making 
use  of  the  fact  that  ea>t  yn  are  a  solution,  it  is  found  that 

n 

tfu+tiV*-  2 

p*l 

If  the  left  members  of  these  equations  are  set  equal  to  zero,  they  become 
precisely  of  the  form  of  the  equations  satisfied  by  the  i/n-  Consequently 
yn=yn  plus  such  particular  integrals  that  the  differential  equations  shall  be 
satisfied  when  the  right  members  are  retained.  The  method  of  finding  the 
particular  integrals  will  be  taken  up  in  §§29-31. 

26.  Construction  of  the  Solutions  when,  for  M  =  0,  the  Roots  of  the  Char- 
acteristic Equation  are  Distinct  and  their  Differences  are  not  Congruent 
to  Zero  mod.  V—  1.  —  The  knowledge  of  the  properties  of  the  solutions  and 
their  expansibility  as  power  series  in  M  leads  to  convenient  methods  for 
constructing  them.  Under  the  conditions  that  for  ju  =  0  the  roots  of  the 
characteristic  equation  are  distinct  and  that  the  difference  of  no  two  of  them 
is  congruent  to  zero  mod.  V—  i,  it  has  been  shown  that  there  are  exactly  n 
distinct  values  of  a  expansible  as  converging  power  series  in  n,  such  that 
xft  =  eattya:  (i  =  l,  .  .  .  ,ri),  where  the  ?/«  are  purely  periodic,  constitute  a 
fundamental  set  of  solutions.  It  will  be  assumed  that  for  M  =  0  no  two  o(" 
are  equal  and  that  the  difference  of  no  two  of  them  is  congruent  to  zero 
mod.  \<r^l,  and  it  will  be  shown  that  the  coefficients  of  the  expansions  of 
the  at  and  ?/«  are  determined,  except  for  a  constant  factor,  by  the  conditions 
that  the  differential  equations  shall  be  satisfied  and  that  the  ytlc  shall  be 
periodic  in  t  with  the  period  2w. 


LINEAR    DIFKKKKXTIAL   EQUATIONS.  37 


r  the  value  of  //u  is  //  =  2  y\>n',  where  the  series  converge  for 


l-o 


all  \n\  sufficiently  small.     It  follows  from  the  periodicity  condition  that 


2  y*  (2ir)M'  =  2 

/-O  *    J.O 


Since  this  relation  is  an  identity,  it  follows  that 


Therefore  each  yu  separately  is  periodic  with  the  period  2*. 

Now  the  original  differential  equations  (92)  after  making  the  transfor- 
mation xl  =  ea'y,  become 


For  /u  =  0  the  roots  of  the  characteristic  equation  belonging  to  these  equations 
are  a*',  a',0)  ,  .  .  .  ,  OL°\  Consider  any  one  of  them,  as  a'".  It  has  been  shown 
that  for  M  7*  0 ,  but  sufficiently  small  in  absolute  value,  at  and  the  yu  are 
expansible  in  converging  series  of  the  form 


Z 

r-O 


(in) 


On  substituting  (111)  in  (110),  arranging  as  power  series  in  p,  and  equating 
coefficients  of  corresponding  powers  in  n ,  there  results  a  series  of  sets  of 
equations  from  which  a»  and  the  yu  can  be  determined  so  that  the  yu  shall 
be  periodic  with  the  period  2ir.  The  determination  is  unique  except  for 
an  arbitrary  constant  factor  of  the  y* .  For  simplicity  of  notation  this 
constant  factor  will  be  determined  so  that  j/"  (0)  =  cu ,  provided  cu  ^  0, 
and  it  can  be  restored  in  the  final  results  by  multiplying  this  particular 
solution  by  an  arbitrary  constant. 

Terms  independent  of  p.    The  terms  of  the  solution  independent  of  n  are 
defined  by  the  differential  equations 


^-1 
the  general  solution  of  which  is 


0-1  .....  »),     (113) 


where  the  V"  are  the  constants  of  integration.  Since  the  y«  are  periodic  with 
the  period  2*-,  and  since,  by  hypothesis,  a^-Cf^O  mod.  V^T,  except  when 
j  =  k,  every  ^  =  Q  if  j?*k.  The  initial  value  of  y™  is  cu;  therefore  iju-1. 


38  PERIODIC   ORBITS. 

If  Cu  were  zero  the  initial  condition  would  be  imposed  upon  another  y(®, 
not  all  of  which  can  be  zero  at  /  =  0.  The  solution  satisfying  the  conditions 
laid  down  is  then 

= 


Coefficients  of  ju.     The  differential  equations  for  the  terms  in  the  first 
power  of  n  are 

(tmX8!/!?-  2  «,,!#  =  -a^+ZflJ"^        (»-l,  .  .  .  ,„).     (115) 
The  general  solution  for  the  terms  homogeneous  in  ?/|"  is 


n                        f«W  «W\I 

V  -«>r     *A    1  *    "                                                                          /I  1ft\ 

*  n  tk  cije  >                                                VAJLOj 
^=1 


where  the  rj(^  are  the  as  yet  undetermined  constants  of  integration,  and  the  ci} 
are  the  same  as  in  (113). 

Using  the  method  of  variation  of  parameters,  we  find 


where  the  (j\"(t)  are  periodic  in  Z  with  the  period  2w.    The  determinant  of 
the  coefficients  of  the  (rjlY  is 


which  can  not  vanish  for  any  finite  value  of  t.     Therefore  the  solutions  of 
equations  (117)  for  (77"*)'  are 


where  the  A(^  are  periodic  functions  of  t  with  the  period  2  IT. 
The  solutions  of  (118)  forjVA;  have  the  form 

,a,  =  ,-«-<>>  p<u  +  J5<o  OVJfe)|  (119) 

where  the  PJ"(0  are  periodic  with  the  period  2ir,  and  the  ££  are  arbitrary 
constants.  For  .7  =  A;  equation  (118)  becomes 

(u£)'  =  AiS>=  -•«'  +  «£,  (120) 

where  8^  is  A£  after  the  terms  -  a^y^  have  been  omitted  from  the  kth 
column.  It  is  a  periodic  function  of  t  with  the  period  2-ir,  and  has  in  general 
a  term  independent  of  t.  It  can  be  written  in  the  form 


where  4"  is  a  constant  and  Q™  (t)  is  a  periodic  function  whose  mean  value 
is  zero.     Then 


LINKAU    DIFKKUKMIAL    Kql'ATIONB.  39 

It  is  clear  that  if  V«  is  to  be  periodic  ihr  right  mcnilicr  (if  this  c(]iiatiun 
must  not  contain  any  constant  terms.  Thrn-foiv 

ol"-*",  (121) 

and 

C-PSI  +  B2,  (122) 

where  P£  is  periodic  with  the  period  2ir  and  B%  is  the  constant  of  integration. 
Upon  substituting  (119)  and  (122)  in  (116),  the  general  solution  with 
the  value  of  a"'  determined  by  (121)  becomes 

•  t*w  «w\i        • 

iff-ZIJQcfciiC  +2i^P«'(0.  (123) 

In  order  that  the  yu  shall  be  periodic  with  the  period  2r,  all  the  B^  must  vanish 
except  Bu.  From  the  condition  that  yu(0)  =  cu  for  all  |M|  sufficiently  small, 
it  follows  that  y"»(0)  =  ctt  and  ^  (0)  =  0  (j  =  1,  .  .  .  oo  ).  From  the  con- 
ditiiin  that  y\"  =  0  at  t  =  0  it  follows  that 


Therefore  the  solution  satisfying  all  the  conditions  is 

1C  =  2  [*,  P%  (0  -  ^Cu  P»  (0)]  .  (124) 

It  remains  to  be  shown  that  the  integration  of  the  coefficients  of  the 
higher  powers  of  n  can  be  effected  in  a  similar  manner.  Let  it  be  supposed 
that  oi",  of,  .  .  .  ,  a*-"  and  the  j&\  y»  ,  .  .  .  ,  t/*-"  have  been  uniquely 
determined  so  that  the  y(£  (t)  are  periodic  with  the  period  2*  and  that  y','  =  0, 
/=!,  ...,m  —  1.  It  will  be  shown  that  the  y™  can  be  determined  so  as 
to  satisfy  the  same  conditions. 

From  equations  (96)  it  is  found  that 


(125) 


Omitting  the  terms  included  under  the  sign  of  summation  with  respect  to  p, 
these  equations  are  identical  in  form  with  equations  (115)  except  for  the 
superscripts  (1)  and  (m).  Obviously  the  integrations  proceed  with  the  index 
(m)  just  as  with  the  index  (1),  and  the  character  of  the  process  is  no  wise 
altered  by  the  inclusion  of  the  terms  under  the  sign  of  summation  with 
respect  to  p,  for  they  are  all  periodic  with  the  period  2*  and  do  not  change 
the  essential  character  of  the  $?(() .  Therefore  a™  and  the  y J"  can  be  uniquely 
determined  so  that  the  y™  shall  satisfy  the  differential  equations  and  be 
periodic  in  t  with  the  period  2r,  and  so  that  at  the  same  time  y™  (0)  =  0. 
The  induction  is  complete  and  the  process  can  be  indefinitely  continued.  The 
solutions  associated  with  the  other  a™'  are  found  in  the  same  way. 


40  PERIODIC    ORBITS. 

27.  Construction  of  the  Solutions  when  the  Difference  of  two  Roots 
of  the  Characteristic  Equation  is  Congruent  to  Zero  mod.  V—i. — It  will 
be  supposed  now  that  a™  — a™  is  congruent  to  zero  mod.  V—  1  and  that  this 
relation  is  not  satisfied  by  any  other  pair  of  af.  The  solutions  associated 
with  af  ,  .  .  .  ,  a(n0>  are  computed  by  the  method  of  §  26  without  modifica- 
tion. It  has  been  shown  that  in  general  a: ,  a2  and  the  ytt ,  yi2  can  be  developed 
as  converging  series  in  integral  powers  of  /z.  It  will  be  assumed  further  that 
the  case  under  consideration  is  not  an  exceptional  one. 

The  general  solutions  of  (96)  for  the  terms  independent  of  n  is  in  this 
case 

"  /V°>   r,w\i 

Iff- *««/•!  ->>'. 

j=i 

Imposing  the  conditions  that  y™  shall  be  periodic  with  the  period  2ir  and 
that  y™  (0)  =  cn ,  these  equations  become,  since  af  —  oi0)  is  an  imaginary  integer, 


where  TJ^  is  so  far  arbitrary. 

Coefficients  of  n-      It  follows  from  (96)  that  the  coefficients  of  /*  must 
satisfy  the  equations 

The  general  solution  of  these  equations  when  their  right  members  are  zero  is 

y(n  =  Sijjfc/*     '     "'  (i=l,  .  .  .  ,  n).      (128) 

On  considering  the  coefficients  rj™  as  functions  of  t  and  imposing  the 
conditions  that  (127)  shall  be  satisfied,  it  is  found  that 

On  substituting  the  values  of  the  yf  from  (126)  and  solving,  there  result 


(129) 


where  the  A™  and  Z)™   are  periodic  functions  of  /  with  the  period  2ir 

/(«)_       (0)\ 

depending  upon  the  fl®  and  e^          ;  .    In  the  first  two  equations  the  unde- 
termined constants  a(,"  and  y™  enter  only  as  they  are  exhibited  explicitly. 


I.INKAK    DIFKKHKMI  \\. 


VI!"V-. 


41 


Kquations  i  l_".i   :nv  ID  he  integrated  ;intl  tin-  results  Mihstituted  in  (128). 
In  order  that  tin-  //     -hall  he  periodic  the  conditions  must  be  imposed  that 


0=-( 

0=-( 
0  = 


n), 


(130) 


whore  6',1,',  &£',  ef,",  ef,1,'  are  the  constant  terms  of  A'/,',  A,",  />,',',  and  D*,"  respec- 
tively, and  where  the  /?",'  are  the  constants  of  integration  obtained  with 
the  last  n  —  2  equations.  These  equations  determine  two  solutions  for  the 
arbitraries  a(,"  and  17™  except  in  those  special  cases  where  the  existence  shows 
the  solutions  are  expansible  in  other  forms. 

Upon  eliminating  V°i  between  the  first  two  equations  of  (130),  it  is 
found  that  a,"  must  satisfy  the  equation 


-  0. 


(131) 


If  the  discriminant  of  this  quadratic  is  not  zero  the  case  is  that  in  the  exist- 
ence proof,  equations  (108),  where  6,  and  b3  are  distinct.  In  this  case,  which 
may  be  regarded  as  the  general  one,  the  solutions  proceed  according  to 
integral  powers  of  n.  If  the  discriminant  is  zero  the  character  of  the  solutions 
depends  upon  the  coefficients  of  terms  of  higher  degree,  and  they  may  proceed 
according  to  powers  of  n  or  =*=  Vji.  It  will  be  supposed  that  the  discrimi- 
nant is  distinct  from  zero,  and  the  method  of  constructing  the  solutions  will 
be  developed. 

Choosing  one  of  the  pairs  of  values  of  a"'  and  ijJJ  which  satisfy  (130),  it 
will  be  shown  that  henceforth  the  solution  is  unique.  Upon  imposing  the 
condition  that  j/',1,'  (0)  =  0 ,  integrating  (129),  substituting  the  results  in 
(128),  and  determining  the  constants  of  integration  so  that  the  solution 
shall  be  periodic,  it  is  found  that 


(132) 


where  #*,',  is  an  undetermined  constant,  and  the  P^'  are  entirely  known 
periodic  functions  of  t,  having  the  jxriod  2r. 


Coefficients  of  p.  The  coefficients  of  ft2  are  defined  by 

*  » 

!_-«»«/*>      v  n   i,(t>  —      /.(Z)i,<0)_/,o>«/1>_u  y  Iff  iim4-flfl) 

ra,  ytl  —  2,  a(>j/,,  —  —a,  j/(,  —a,  yn  -\-  L  \v\,  yn-rvt, 


(133) 


42 


PEEIODIC    ORBITS. 


The  general  solution  of  these  equations  when  the  right  members  are  neglected 
is  the  same  as  (128)  except  that  the  superscripts  are  (2)  instead  of  (1). 
On  varying  the  if®,  the  equations  corresponding  to  (129)  are 


(134) 


The  undetermined  constants  af  and  fi^i  are  exhibited  explicitly  in  the  first 
two  equations,  and  it  is  to  be  noted  that  A",*  and  A^'  are  precisely  the  same 
functions  of  t  as  those  which  appeared  in  (129). 

In  order  that  these  equations  shall  lead  to  periodic  values  of  the  y™ 
the  undetermined  constants  must  satisfy  the  conditions 


(135) 


0=  -  af>    l  -„» 
0=  -aft«-a 


(j=3,  .  .  .  ,  ri), 


The  first  two  equations  are  linear  in  af  and  &£,  and  they  determine  these 
quantities  uniquely  unless  their  determinant  is  zero.     The  determinant  is 


A=- 


-«» 


(!) 


On  eliminating  77^  and  a™  by  means  of  (130),  it  is  found  that 


where  D  is  the  discriminant  of  (131).  Since  by  hypothesis  D  is  not  zero, 
the  determinant  A  is  not  zero.  Hence  af  and  B%  are  uniquely  determined 
by  (135).  Having  determined  B(£  and  af,  equations  (134)  are  integrated 
and  the  results  are  substituted  in  the  equations  corresponding  to  (128). 
Then  the  conditions  that  y^  (0)  =  0  are  imposed  and  the  final  solution  at 
this  step  becomes 


(136) 


where  B%  is  undetermined  until  the  next  step  of  the  integration. 


LINKAIt    IHKI  KKKVtUL   EQUATIONS.  -M 

The  next  step  is  similar  to  the  preceding  and  all  the  equations  are  the 
same  except  the  superscripts  are  (3)  and  (2)  in  place  of  (2)  and  (1)  respec- 
tively. The  determinant  of  the  equations  corresponding  to  (135)  is  precisely 
the  same.  In  fact  all  succeeding  steps  are  the  same,  and  the  whole  process 
can  he  repeated  as  many  times  as  is  desired.  The  solutions  associated  with 
ajj",  ....  a||"  are  found  as  they  were  in  §2i>. 

For  congruence-  of  higher  order  similar  methods  can  be  used,  and  in 
the  cases  which  are  exceptions  to  this  mode  of  treatment  the  existence 
discussion  furnishes  a  sure  guide  for  the  construction  of  the  solutions. 

28.  Construction  of  the  Solutions  when  two  Roots  of  the  Characteristic 
Equation  are  Equal.  —  It  will  he  supposed  aij"  =  aj"  and  that  all  the 
remaining  a'°'  are  mutually  distinct  and  distinct  from  a™.  The  solutions 
depending  upon  aJ0>  ,  .  .  .  ,  a'.0'  can  be  computed  by  the  method  of  §26.  It 
has  been  shown  that  the  two  solutions  proceeding  from  a',0)  are  in  general 
expansible  as  power  series  in  -N/M.  The  detailed  discussion  will  be  made 
only  for  the  general  case,  where 


(137) 


Terms  independent  oj  n.     The  terms  independent  of  n  are  defined  by 

W'+tt-*fdff-Q          (,•=!,...,  n).     (138) 
The  general  solution  of  these  equations  is 

-"'")'.  (139) 


In  order  that  the  (/;"'  shall  be  i>eriodic  with  the  period  2r  and  the  initial  value 
Cn  of  if™  shall  be  obtained,  the  V*  must  satisfy  the  conditions 


The  solution  satisfying  all  the  conditions  is  then 

t/(,?  =  cn  (i-l  ----  ,»).     (HO) 

Coefficients  of  n*  .     The  coefficients  of  ^  are  defined  by  the  equations 


44  PERIODIC    ORBITS. 

On  neglecting  the  right  members,  the  general  solution  of  these  equations  is 
l£  =  i^i+i?$?fc,,+fcn)  +  2  tfc^-^'.  (142) 

j=3 

The  method  of  variation  of  parameters  leads  to  the  conditions 

(Wcn  +  (r,%Y(ct2+tcn)+  2  WA/**"^'  =  -a'X         0-1,  .  .  .  ,  n). 

j  =  3 

On  solving  these  equations  for  the  (r?™)',  it  is  found  that 

OlS)'=-a?),  G$)'  =  0  (j  =  2,  ...,n). 

Consequently 

,«=*S-aS%  „»  =  £«  (j  =  2,  ...,n).      (143) 

On  substituting  the  values  from  (143)  in  (142),  the  result  becomes 


To  satisfy  the  conditions  for  periodicity  and  to  make  T/,"  (0)  =  0,  the  Bn 
must  fulfill  the  relations 

B(tl  =  a™,         B(ilcn-\-B(2lc12  =  0,          -B(/i=0          C?=3,  .  .  .  ,n).      (144) 
Then  the  solutions  satisfying  all  the  conditions  become 

Vn  =  (-  fctl+c^  ai",  (145) 

where  the  constant  a"'  remains  as  yet  undetermined.  It  is  to  be  observed  that 
not  all  the  coefficients  of  a"'  can  vanish,  for  otherwise  the  determinant  |cj 
itself  would  vanish. 

Coefficients  of  /z.    The  coefficients  of  /i  are  determined  by  the  equations 

n  n 

(2)  „  (0)  (1)     (1)    I      ^    /l(l)      (0)  /  •        -t  \  /I  Ad\ 

=  ~a\  Mi\  ~ai  2/n+  2  ^uUn         (»«•!,...,  n).     (14b; 


The  solution  of  the  homogeneous  terms  is  of  the  same  form  as  (142),  and  by 
varying  the  constants  of  integration,  it  is  found  that 


2 


(147) 


LINEAR    DIKKKUKNTIAL   EQUATIONS. 

The  solutions  of  these  equations  for  the  (a'/,')'  arc 


45 


(148) 


where  the  &%(i)  and  />™(0  an-  known  jwriodic  functions  of  t.     The  first 
equation  gives  rise  to  integrals  of  the  type 

—  a,(  t  *^jtdl=  +  ^  gfajt+^^jt' 
J  3 

The  second  equation  gives  rise  to  the  corresponding  integral 


When  these  results  are  substituted  in  equations  (142)  the  terms  of  the 
type  t  ^jt  destroy  each  other.    Hence  at  this  step 


, 


(149) 


where  the  P%(t)  are  periodic  functions  of  /,  and  6"'  and  <?*  are  the  constant 
terms  in  A"'  and  Z)^.  Equations  (149)  are  the  general  solutions  of  equa- 
tions (147).  In  order  to  satisfy  the  conditions  for  periodicity  and  the 
initial  condition  j/'n  (0)  =0,  the  constants  aju  and  B<%  must  fulfill  the  relations 


(150) 


(J-3 ,»). 


The  constant  ojl)  still  remains  undetermined.    The  solutions  now  arc 


<0,  (151) 

where  the** are  known  periodic  functions  of  <  having  the  period  2ir.  After 
making  a  choice  as  to  a',",  provided  6™^0,  it  is  found  that  a(,Z)  is  determined 
uniquely  by  the  periodicity  conditions  for  y™.  The  process  can  be  con- 
tinued indefinitely  and  the  constants  are  determined  uniquely.  The  other 
solution  associated  with  a',01  is  obtained  by  taking  the  other  determination 
of  a',",  and  the  solutions  depending  on  aj",  .  .  .  ,  ajj"  by  the  method  of  §  26. 
The  chief  types  of  cases  have  been  treated,  and  the  exceptions  to  them  are 
developed  similarly  according  to  the  forms  indicated  in  the  existence  proofs. 


46  PERIODIC    ORBITS. 

IV.    NON-HOMOGENEOUS  LINEAR  DIFFERENTIAL  EQUATIONS. 

29.  Case  where  the  Right  Members  are  Periodic  with  the  Period 
2ir  and  the  a,  are  Distinct.  —  Take  the  set  of  differential  equations 

x't-  2et)xJ  =  g((t)  (i=\,  .  .  .  ,n),        (152) 

where  the  6tJ  and  the  </,(£)  are  periodic  in  t  with  the  period  2ir.  For  the  left 
members  set  equal  to  zero  the  form  of  the  solution  of  (152)  is 

x(=  Zv^'ytj  ,  (153) 

where  the  77^  are  arbitrary  constants,  the  a,  are  the  characteristic  exponents 
which  are  supposed  distinct,  and  the  ytl  are  periodic  in  t  with  the  period  27r. 
By  the  method  of  variation  of  parameters,  it  is  found  that 

Z^^-ft®.  (154) 

The  determinant  of  the  coefficients  of  the  ^  is  the  determinant  of  the  funda- 
mental set  of  solutions.  Since  the  6fj  are  assumed  to  be  regular  for  all  finite 
values  of  t,  it  follows  from  §18  that  this  determinant  can  not  vanish  for  any 
finite  value  of  t.  This  determinant  is 

n 

2    Cijt 

Ae"     , 

where  A  is  the  determinant  of  the  ytj.  If  A^  denotes  that  which  A  becomes 
when  the  jth  column  is  replaced  by  the  gt  (f),  the  solutions  of  (154)  for  the  ^  are 

Hi-  %  «-"",  (155) 

and  consequently 

(156) 


The  quotient  A,/  A  is  a  periodic  function  of  t,  continuous  and  finite  in  the 
interval  Q^t^2ir.     Therefore  it  can  be  expanded  into  the  Fourier  series 


^  =  0^+2   [o|?  cos  w<  +  V*  sin  mt] . 
If  aJ+fft^O  (j  =  l,  .  .  .  ,  n;  m  =  l,  .  .  .  ,  oo),  the  integral  becomes 

cos  mt-\ -™ — —^sin  wif    (157) 

a^+w2  J 


a«fm 

so  that  r 


where  theP,(<)  are  periodic  with  the  period  2ir,  and  the  B}  arc  constants  of 
integration.  On  substituting  these  values  of  the  77,  in  (153),  the  general 
solutions  of  (152)  become 

x(=      £,«•<%+  iW«M      (i=l,  .  .  -  ,  n).     (158) 


PARTICULAR   SOLUTIONS   OK    I.INKAR   EQUATIONS.  47 

Now  suppose  a.  =  k  \  -  i  .  wliere  k  is  an  integer.     Then  the  term 

JV'V~  '[at  cos  kt+bt  sin  kt]  dt 
become-.  after  the  integration  ha-  been  carried  out, 

|  (a,  -  bt  v^  i  )  t  +  Jg  (a.v^rf  -  &,)  (cos2to  -  v^=T  sin  2  Ar<)  . 
Therefore  the  expression  corresponding  to  (158)  becomes  in  this  case 

z,=  Zi£/wy<,  +  F1(0  +  J(a.-^V^*ea''y«,  (159) 

where  the  P\(t)  are  periodic  with  the  period  2ir. 

Therefore,  if  the  characteristic  exponents  are  distinct  and  none  of  lliem  is 
congruent  to  zero  mod.  V  —  l,  and  if  the  gt(t)  are  periodic  with  the  period  2r, 
tlt>  n  the  particular  integrals  are  also  periodic  with  the  period  2r.  But  if  some 
of  the  characteristic  exponents  are  congruent  to  zero  mod.  V—\,  then  the  par- 
ticular integrals  in  general  contain,  in  addition  to  periodic  terms,  the  corre- 
sponding part*  of  the  complementary  function  multiplied  by  a  constant  limes  t. 

30.  Case  where  the  Right  Members  are  Periodic  Terms  Multiplied  by 
an  Exponential,  and  the  a,  are  Distinct.—  Consider  the  case  where  the  0,(0 
have  the  form 


the  /4(0  being  periodic  with  the  period  2r.     When  X  =  I  \/~—  i  is  a  pure 
imaginary,  this  form  includes  such  cases  as 

gt(t)  =  X[ 


If  the  differential  equations,  which  are  now  of  the  form 

l-jtf&i-ffM,  (160) 

are  transformed  by  xt  =  e*z{  ,  they  become 

(161) 


and  have  the  same  character  as  those  treated  in  §29.  If  the  character- 
istic exponents  a.,  of  (160)  are  distinct,  then  the  characteristic  exponents 
of  (161)  are  a,-X,  and  are  also  distinct.  Applying  the  results  of  the  pre- 
ceding case,  it  is  seen  that  if  no  a,—  X  is  congruent  to  zero  mod.  v'=T,  then 
the  solutions  of  (161)  are 


where  the  Qt(t)  are  periodic  with  the  period  2*.     Therefore  the  solutions  of 
(160)  are 

(»'-i  ----  ,»)•     (162) 


48  PERIODIC    ORBITS. 

But  if  one  of  the  a,,  say  at,  is  congruent  to  X  mod.  V-7!  ,  then  the  z, 
have  the  form 


and  therefore  the  expressions  for  the  xt  become 

t  (f)  +ctt  ea"'yit  .  (163) 


These  results  may  be  stated  as  follows:  //  the  gt(t)  have  the  form 
0i  (0  "  «**/«(*)  »  where  ft(t)  =ft(t  +  2?r),  and  if  none  of  the  characteristic 
exponents  is  congruent  to  X  mod.  V—l,  then  the  particular  solution  has  the  form 

xt  =  e*Qt(t)  (i=l,...,n), 

where  the  Q((f)  are  periodic  with  the  period  2-w;  but  if  one  of  the  characteristic 
exponents,  at,  is  congruent  to  \  mod.  V—l,  then  the  particular  solution  has 
the  form 

z,  =  ewQ,(0+c**ea*V«  (t  =  l,  .  .  .  ,«), 

where  the  Qt(()  are  periodic  with  the  period  2  IT. 

3  1  .  Case  where  two  Characteristic  Exponents  are  Equal  and  the  Right 
Members  are  Periodic.  —  Suppose  a2  =  ai.  Then  the  solutions  of 

xl-ZOuX^O  (*-!,...,  n), 

in  general  have  the  form 


yit  (i=  1,  .  .  .  ,  n).      (164) 

For  the  associated  non-homogeneous  equations 

*t-,Z4,*,-ft(<)  (165) 

it  is  found  by  applying  the  method  of  the  variation  of  parameters  that 
ea>'ynr,{+ea>1  (ya+tyM+  S  ea»  yt,-n',  =  gt(t)  (i=l,  ...,»). 

^=3 

On  solving  these  equations  for  the  ^  ,  the  results  are  found  to  be 

tyn),  ya  ,  .  .  .  ,  ytn\e-a>t, 
gf(t),  yt3,  .  .  .  ,  ytn\e-ati, 


where  A  is  the  determinant  \ytj\  .     The  expansions  of  these  determinants 
have  the  form 

=  -w(0-e—<P,(0>  r,'2  =  e-a'T,(t),  } 

O  (j-3,  ...,  n),         I 

where  the  Pf(t)   (j=l,  .  .  .  ,  n)  are  periodic  with  the  period  2ir. 


I'Alilli  I  I.\K   SOLUTIONS   OF   LINEAR   EQUATIONS.  49 

Suppose  no  a,  is  congruent  to  zero  mod.  \/^T.     Then  it  is  easily 
found  that 


(167) 


where  /?,(0,  •  •  •  ,  #»(0  are  periodic  with  the  period  2r.  On  substituting 
these  values  in  (164),  the  solution  become- 

x,  =  £,  ea"y(1  +#aea%w+<  j/n)  +  £5,  ea>«/(,  +  2  fl/<)  y,,. 

• 

The  1'  A*//,  are  periodic  with  the  period  2r.     Hence.  if  tiro  of  the  characteristic 
j-i 

exponent*  HIT  eijinil  hut  mine  of  them  is  congruent  to  zero  mod.  V^l,  then  the 
/Hirtieiilnr  .Dilution  also  /.s  /><  riutlic  irith  the  period  2ir. 

The  ease  where  one  a,  (j  =  3  ,  .  .  .  ,  /<)  is  congruent  to  zero  mod.  V—l 
is  a  combination  of  the  present  ease  with  the  second  part  of  that  treated  in 
S'J'.i.  and  that  where  a,  =  a,  is  congruent  to  zero  mod.  V—i  does  not  differ 
in  any  essentials  from  that  where  a,  =  a,  =  0. 

\<>w  suppose  a,  =  al  =  0.  Then  the  equations  which  correspond  to 
(166)  become 


The  Pj(t)  are  periodic  with  the  period  2r  and  can  be  written  in  the  form 

00 

Pj  =  a,  +  2  [atj  cos  kt+bu  sin  kt\. 
Hence  the  TJ,  are  found,  by  integrating,  to  have  the  form 


(j=3 
where 


These  values  substituted  in  (164)  give  for  the  complete  solutions 

[alt+la1(']  yll+aitya+ 


Hence,  irh<  n  the  gt(t)  are  periodic  with  the,  period  2*,  and  when  two  of  the  char- 
acteristic exponent*  arc  not  only  equal,  but  are  zero,  then  the  particular  intei/rul 
int'olris  nut  null/  t  hut.  in  </<  neral,  alto  f  outside  of  the  trigonometric  symbols. 
Of  course  it  may  happen  that  a,  and  at  are  either  or  both  zero. 

Those  particular  cases  have  been  treated  which  will  be  most  useful  in 
in  the  applications  which  follow.  Any  others  which  may  arise  can  be 
discu—  rd  in  a  similar  wa\  . 


50  PERIODIC   ORBITS. 

V.    EQUATIONS  OF  VARIATION  AND  THEIR  CHARACTERISTIC 

EXPONENTS.* 

32.  The  Equations  of  Variations.  —  The  preceding  considerations  find 
immediate  application  in  dynamics  in  the  study  of  small  variations  from 
known  periodic  solutions.  Suppose  there  arc  given  the  equations 

^=Xt  (»  =  !,...,  n),  (168) 

where  the  Xt  are  functions  of  the  xt,  and  that  they  are  satisfied  by 


the  <pt(t)  being  periodic  functions  of  t  with  the  period  2ic.     These  are  called 
the  generating  solutions. 

Let  the  initial  conditions  be  varied  slightly  and  put 

*,(0)=^(0)+ft,  (170) 

where  the  ft  are  small  arbitrary  constants.    The  value  of  the  xt  for  any  t  will  be 


the  £,  being  functions  of  t  which  for  at  least  a  short  interval  of  time  will 
remain  small.  On  substituting  (171)  in  (168)  and  expanding  the  right 
members  as  power  series  in  the  £,  ,  it  is  found  that 


Tt  =  ?  |f  *>  +  higher  degree  terms  (t-1,  ...,*),      (172) 

the  xi  being  replaced  by  <pt(t)  in  the  partial  derivatives.  Since  the  <f>t(t)  are 
periodic  in  t,  so  also  are  all  the  coefficients  of  (172).  The  &  are  expansible  as 
power  series  in  theft  which,  by  the  Cauchy-Poincare"  theorem,  §  16,  converge 
for  any  preassigned  interval  of  time  provided  the  |ft|  are  sufficiently  small. 
The  differential  equations  for  the  linear  terms  in  the  ft  are  the  linear 
terms  of  (172),  or 


These  equations  are  known  as  the  equations  of  variation. 

Suppose  the  solution  (169)  contains  an  arbitrary  constant  c,  that  is, 
one  not  contained  in  the  differential  equations  (168).  If  c  =  c0-\-y,  the  <p((0 
are  expansible  as  power  series  in  7  of  the  form 


Obviously 

=  d^'^J.1aVr 

dc2 


*Thc  subject  of  this  section  and  other  related  questions  have  been  treated  by  Poincare,  Les  Methodes 
Nou'jclles  <k  la,  Mecanique  Celeste,  vol.  I,  chap.  4. 


KQI'AlloV-    til-     \\I{IATI<>\     AM)    lH\K\i    IKKISTlr    KXl'iiNKNTO.  51 

is  a  solution  of  equation-    I  7_'     and  consequently 

£,=  (i-1, »)  (174) 

is  a  solution  of  equations  t  1~:{). 

One  such  constant  i-  always  present  when  the  A,  do  not  contain  t 
explicitly  and  the  v~, \h  are  not  mere  constants,  for  then  the  origin  of  time  is 
arbitrary.  Hence  in  this  case 

£ d<pt  _       ''y'  (\7^\ 

is  a  solution  of  (17:*i.  Another  such  constant  usually  present  is  the  scale 
factor  which  determines  the  si/oof  the  generating  orbit.  If  there  are  p  such 
arbitrary  constants  in  the  generating  solution,  the  equations  of  variation 
have  /i  solutions  of  the  form  (174). 

If  the  equations  (168)  admit  an  integral  which  is  independent  of  t, 

FI  (*  i  *»)  =  <"i » 

where  r,  is  an  arbitrary  con-taut,  the  .r,  can  be  replaced  by  <pt(t)  +  tt  in  the 
integral  and  the  integral  can  be  expanded  in  powers  of  {, .  The  result  is 

df>i*   i  a//'^  ,  dl<\. 

?=  a     £i  +  T~«~r"  •  •  •  +T~I{»+ higher  degree  terms. 

The  constant  7  is  a  power  series  in  theft,  and  therefore  the  linear  terms  are 

- 1  /  /          1 1.'  '  1 1.' 

*f    J^  '  I-    J^  _L  1  t  flTfi1^ 

which  is  therefore  an  integral  of  equations  (173).  The  coefficients  db\/dxt  are 
I>eriodic  functions  of  /,  the  x,  having  been  replaced  by  <pt(t)  after  differentiation. 

33.  Theorems  on  the  Characteristic  Exponents. —The  existence  of 
arbitrary  constants  in  the  generating  solutions  and  the  existence  of  integrals 
of  equations  (168)  have  an  intimate  connection  with  the  characteristic 
exponents  of  the  solutions.  These  solutions  arc  in  general  of  the  form 

^  =  e(M/,(0  li-l,  •  •  -  ,n),        (177) 

the/,  being  j>eriodic  with  the  period  2ir.  The  solutions  (175)  have  the  form 
(177),  but  since  the  tpt  are  periodic  so  also  are  their  derivatives,  and  the 
characteristic  exponent  of  this  solution  is  zero.  There  is  an  exception  only 
if  the  ^>,  are  constants,  in  which  case  the  solution  (175)  disappears. 

The  solution  obtained  by  differentiating  with  respect  to  the  scale 
constant,  which  will  be  denoted  by  a,  will,  in  general,  have  the  form 


^,  and  \jft  being  periodic.  The  characteristic  exponent  of  this  solution  is 
zero.  If  the  generating  solutions  have  p  distinct  arbitrary  constants,  the  equa- 
tions of  variation  will  have  at  least  p  characteristic  exponents  equal  to  zero. 


52  PERIODIC    ORBITS. 

From  the  existence  of  the  integral  (176)  it  follows  also  that  at  least  one 
of  the  characteristic  exponents  is  zero;  for  all  solutions  have  the  form 


and  substituting  them  successively  with  respect  to  the  index  j  in  (176),  we  get 

-^V,"  (j  =  i,...,n).         (178) 


The  left  members  of  these  equations  are  periodic  with  the  period  2ir,  except, 
perhaps,  for  coefficients  which  are  polynomials  in  t.  It  follows,  therefore, 
that  either  the  a_,=0  mod  V^l,  or  all  the  7</)  =  0.  In  this  connection  a 
congruence  has  the  same  properties  as  an  equality,  and  they  need  not  be 
distinguished  from  each  other.  If  all  the  a}  are  distinct  from  zero,  then 
<)  =  = 


7    =         =   ,  .  .  .  ,  ri)  and  (178)  becomes 


(179) 


Since  the  determinant  |  ftj  \  T*  0,  these  equations  can  be  satisfied  only  if 


Therefore,  unless  the  integral  (176)  vanishes  identically  at  least  one  of  the 
characteristic  exponents  is  zero.  Suppose  that  a,  =  0  and  that  7"'  ^  0. 
It  is  possible  then  to  solve  the  equations  corresponding  to  (178)  uniquely 
for  the  dFj/dz,  in  terms  of  7"'  and  the  f(j  . 

Suppose  now  there  is  a  second  integral  F2  (xl  ,  .  .  .  ,  xn)  =  c2  .     Then 

=  const. 
t 

On  substituting  in  this  equation  successively  the  n  fundamental  solutions 
for  the  £,  it  follows,  since  at  =  0,  that 

0'  =  2,  ...,n).  (180) 


If  0,^0,  and  therefore  5^  =  0  (j=2,  .  .  .  ,  n),  these  equations  can  be 
solved  uniquely  for  dF2/do;,  in  terms  of  /„  and  5™.  It  results  that,  aside 
from  a  constant  factor, 

dF,_dF1 

dxi  ~  dxt  ' 

and  the  second  integral  is  identical  with  the  first.  But  if  Fl  and  F2  are 
distinct,  then  there  must  be  at  least  two  characteristic  exponents,  say  c^  and 
Oj  ,  which  are  zero.  Proceeding  in  this  manner  it  follows  that  if  the  equa- 
tions of  variation  admit  of  p  linearly  distinct  integrals  not  identically  zero, 
then  there  are  p  characteristic  exponents  equal  to  zero. 


EQUATIONS   OF   VARIATION    AM)    CHARACTERISE'    KXI'i  INKNT8.  53 

If  the  original  differential  equations  have  the  form 

He   ~~  dx,  ('"1 n)> 

which  is  the  case  usually  in  celestial  mechanics,  they  may  be  reduced  to 
equations  involving  only  first  derivatives  by  writing 


If  the  generating  solution  is 


and  the  equations  of  variation  are  formed  by  putting 


there  will  result 

&. 

<"  (181) 


- 
r 


rf/       dz,  dit        dxt  d*  '  3x,  dx.  •  ' 

The  main  diagonal  of  the  right  members  of  these  equations  (considered  as 
a  determinant  matrix)  contains  only  zero  elements.  Therefore,  by  §18,  the 
determinant  of  any  fundamental  set  of  solutions  of  these  equations  is  a 
constant.  But  the  determinant  of  the  fundamental  set  of  solutions 


a/i 


/ 

has  the  form  A  =  e'~  P  (t)  .  This  must  therefore  be  a  constant,  from  which 
it  follows  that  the  sum  of  the  characteristic  exponents  is  zero  since  P(t)  has 
the  period  2*. 

Suppose  £"',  TjJ"  and  #?,  jjf  (t  =  l,  ...,n)  are  any  two  solutions  of 
equations  (181).    Then 


"5 
and  also 


«<»  £<"  i  un  . 

"        "  ^' 


dj?     d'v  ^  .          ^  .  ^ 

""^'        f$-- 


From  these  equations  it  follows  that 

S(j»d$>      mdtf\_Q  *?  («»di?_p4i? 

k*    dt      *l~dt)~    '  2*\*>    dt      *   dt 

The  sum  of  these  two  equations  is 

s[(^f+^D-(<f+^f  )]=<>• 

which  can  be  written  .  . 


54  PERIODIC    ORBITS. 

Consequently 


2  (W-SfY/0^  const.  (186) 


1=1 


The  relation  (186)  between  any  two  solutions  leads  to  important  conclu- 
sions respecting  the  characteristic  exponents.  Suppose  the  ^J)  and  ^'  are 

£"  =  eai'fti(t),         ^  =  eaitgtj(t}  (i=i,  .  .  .  ,n;/-l,  .  .  .  ,2/0, 

where  fi}  and  gtj  are  polynomials  in  i  with  periodic  coefficients,  and  that  they 
constitute  a  fundamental  set  of  solutions.  On  substituting  any  two  of  these 
solutions  in  (186)  and  dividing  through  by  the  exponential,  there  results 

n 

S(  -f      f.  /     n     \  n~ (Cij-{-O,k)t  /I  Qf7\ 

\J  ij ,fik       Jikaij)  * —    Ijk  "  *  \     ^ '  / 

It  follows  from  the  character  of  the  left  member  of  this  equation  that  either 
ei^+aj.  =  0,  or  7^  =  0.  It  will  be  shown,  however,  that  7;t  can  not  be  zero 
for  every  k. 

Suppose  j  is  kept  fixed  and  give  to  k  all  the  values  from  1,  .  .  .  ,  2n. 
Suppose  7,4  =  0  (k  =  l,  .  .  .  ,  In).  Then  one  equation  of  (187)  is  an 
identity  and  the  others  are  linear  in  the  ftj  and  the  gtj .  The  determinant 
of  this  set  of  linear  equations  is  the  determinant  of  the  fundamental  set 
and  is  not  zero.  Hence  they  can  be  satisfied  only  by  ftj=gtj=0-  But 
this  also  is  impossible  since  ftj  and  gtj  are  a  solution  of  the  fundamental  set. 
Therefore  not  all  the  yjf  can  be  zero.  Hence  for  some  k 

^j~r  &„  ~  0,  7j*72^0.  (18o) 

But  since  a,  is  any  one  of  the  characteristic  exponents,  it  follows  that  corre- 
sponding to  each  characteristic  exponent  there  is  another  one  which  differs 
from  it  only  in  sign. 

If  two  of  the  a,  are  equal  but  not  equal  to  zero,  then  there  are  two  others 
which  are  also  equal  and  which  differ  from  the  first  two  only  in  sign.  In 
order  to  show  this  suppose  a_,  =  aj+,  =  —  am.  Then  aM  =  am+I ,  because  from 
(188)  it  follows  that  o,+am  =  0,  7jm^0.  If  7;t  =  0  (k  =  l,  .  .  .  ,  2n,  k^m), 
then  (187)  can  be  solved  for  ftj  and  gtj  uniquely  in  terms  of  f,k  and  g,k 
(k  =  l  .  .  .  ,  In).  Now  the  corresponding  equations  for  /w+1  and  gt,j+1 
will  differ  from  (187)  only  in  that  j  is  replaced  by  j+1.  They  can  be 
solved  uniquely  for  /4iJ+1  and  </JiJ+1 ,  but  this  solution  will  differ  from  the 
solution  for  ftj  and  gfj  only  by  a  constant  factor.  Since  this  is  impossible 
it  follows  that  7;+i,m+1^0,  and  consequently  aj+l+am+1  =  0.  In  the  same 
manner  it  can  be  shown  that  if  p  of  the  a,  are  equal,  then  p  other  a;  are 
also  equal  and  differ  from  the  first  set  only  in  sign. 


CHAPTER  II. 

ELLIPTIC  MOTION.* 

34.  The  Differential  Equations  of  Motion.  —  Consider  two  spheres 
whose  materials  are  arranged  in  homogeneous  spherical  layers  concentric 
with  their  centers.  Then  they  attract  each  other  as  material  points,  their 
orbits  are  plane  curves,  and  the  differential  equations  which  the  motion  of 
one  relative  to  the  other  must  satisfy  are,  in  i>olar  coordinates, 


In  writing  these  equations  the  origin  has  been  placed  at  one  of  the  bodies 
and  the  variables  r  and  v  are  measured  in  the  plane  of  motion. 

Equations  (1)  are  easily  integrated,  and  the  integrals  show  that  the 
relative  motion  is  in  a  conic  section  for  any  initial  conditions.  If  the  initial 
velocity  is  not  too  great  the  orbit  is  an  ellipse,  and  the  discussion  will  be 
limited  to  this  case.  While  the  ordinary  integration  of  (1)  shows  that  under 
certain  conditions  the  orbits  are  ellipses,  it  does  not  express  the  coordinates 
explicitly  in  terms  of  the  time.  The  explicit  developments  are  obtained 
through  solving  Kepler's  equation,  generally  by  Lagrange's  method  or  by 
means  of  Bessel's  functions.  In  treating  elliptic  motion  as  periodic  motion 
the  expressions  for  r  and  v  in  terms  of  t  will  be  derived  directly  from  the 
differential  equations. 

On  integrating  the  second  equation  of  (1),  and  by  means  of  this  integral 
eliminating  dv/dt  from  the  first,  it  is  found  that 

,dv  <Fr     c*      fc'C 


Assume  that  the  conditions  for  an  elliptical  orbit  are  satisfied,  and  let 

a  =  the  major  semi-axis  of  the  orbit; 

e  =  the  eccentricity  of  the  orbit; 

w  =  the  mean  angular  velocity  in  the  orbit; 

T  =  the  time  the  body  passes  its  nearest  apse; 

T  =  w(t  —  T)  =  the  mean  anomaly. 

It  is  found  from  the  integrals  of  (2)  thatf 

q-O-MVa-e').        (3) 


•This  chapter  was  written  in  1900  and  a  brief  account  of  it  was  published  in  the  Astronomical  Journal 
Vol.  XXV,  May,  1907. 

tMoulton'g  Introduction  to  CeUttial  Mechanic*,  pp.  173-8. 


56  PERIODIC    ORBITS. 

On  making  use  of  (3)  and  using  r  as  the  independent  variable,  the 
second  equation  of  (2)  becomes 


dV 


,       _Q 

h    2 


dr2  r3  r 

Equation  (4)  is  satisfied  by  the  circular  solution  r  =  a.  Let  the  radius  in 
the  elliptic  orbit  be 

r  =  a(l-pe),  (5) 

where  p  =  1,  dp/dr  =  Q,  at  r  =  0.  At  the  half  period  T  =  TT,  r  =  a(l+e).  There- 
fore p  =  —  1  at  T  =  7T.  These  are  the  extreme  values  of  r  in  elliptical  motion, 
and  therefore  +  l!>p^  —1. 

Upon  substituting  (5)  in  (4),  the  latter  becomes 

,      p-e     _Q  (6) 

2       (1  —  pe)3 

The  second  term  in  this  equation  can  be  expanded  as  a  power  series  in  e  for 
all  the  values  of  p  if  e\  <  1,  as  is  explicitly  assumed,  giving 


(t  +  l)[t-(i+2)PV-V,  (7) 

and  the  first  equation  of  (2)  becomes  by  the  same  substitutions 

(8) 


<=0 


35.  Form  of  the  Solution. — The  solution  of  equation  (7)  will  first  be 
considered.  After  it  has  been  found,  v  is  determined  from  (8)  by  a  simple 
quadrature. 

Equation  (7)  belongs  to  the  type  treated  in  §§14-16,  and  therefore 
can  be  integrated  as  a  power  series  in  e,  and  \e\  can  be  taken  so  small  that 
the  series  will  converge  for  0  ^  r  ^  2ir.  Since  the  body  moves  so  that  the  law 
of  areas  is  satisfied  and  completes  a  revolution  in  2ir,  p  is  periodic  with  the 
period  2ir.  Consequently,  if  the  series  converges  for  O^T  5=271-,  it  converges 
for  all  real  values  of  T.  It  is,  indeed,  possible  to  find  the  precise  limits  for  \e\ 
within  which  the  scries  will  converge  for  all  values  of  T,  and  outside  of  which 
they  will  diverge  for  some  values  of  r.  The  problem  was  first  solved  by 
Laplace,*  who  found  that  the  series  converge  for  all  T  if  e<  0.6627  .  .  .  , 
which  is  far  above  the  eccentricity  of  the  orbit  of  any  planet  or  satellite  in 
the  solar  system. 

*Mecaniqtie  Celeste,  vol.  V,  Supplement;  see  also  Tisserand's  Mecanique  Celeste,  vol.  I,  chapter  16, 
and  a  demonstration  by  Hermite,  Courts  a  la  Fac.  des  »Sa.  de  Paris,  3d  edition  (1886),  p.  167. 


Kl. Ml'TIC    MOTION.  57 

The  solution  of  (7.)  can  !><•  written  in  the  form 

P=2P>(ry,  (9) 

J-O 

where  the  p,(T)  are  functions  of  T.  According  to  §  1;~>  and  the  initial  condi- 
tions, the  constants  of  integration  which  arise  are  to  be  determined  by  tlie 
conditions 

=  l,p,(0)=0      (f-i,       .00);      ^(0)=0      (i-1,  .  .  .  oo).      (10) 


As  p  is  ]M-riodic  with  the  period  2r,  it  follows  that  p(r+2ir)=p(T);  whence 

=p,(r)e'.  (11) 


Since  (11)  is  an  identity  in  e  it  follows  that  p,(T+2T)ssp,(T).  But  this  is 
simply  the  definition  of  periodicity.  Therefore  each  p,  separately  is  periodic. 

The  body  is  at  its  nearest  apse  when  T  =  0,  and  the  orbit  is  symmetrical 
with  respect  to  the  line  of  apses.  Therefore  it  follows  that  p  is  an  even 
function  of  T.  Since  p  is  periodic  in  r  identically  with  respect  to  e,  each 
P,(T)  is  expressible  as  a  sum  of  cosines  of  integral  multiples  of  r. 

If  the  sign  of  e  in  (6)  is  changed,  then  the  body  is  at  its  farthest  apse 
when  T  =  0.  Consequently  changing  the  sign  of  e  and  increasing  T  by  T 
does  not  change  the  value  of  r.  Since  r  =  a(l  —  pe)  ,  it  follows  that 

-e)'-1.  (12) 


Therefore  when./  is  even,  P,(T)  involves  cosines  of  only  odd  multiples  of  T; 
and  when  j  is  odd,  p,(r)  involves  cosines  of  only  even  multiples  of  r. 
If  we  substitute  (9)  in  (8),  we  get 


(13) 


;-o 


d»        ,. 
d'r=Vl-> 


where 

•  * 

C,t  .....  ,.  -.»•-.,,         (t,+H+  •  •  •  +it-i).     (14) 
(|  .  ij  ....  i,  . 

Suppose  ijt+  •  •  •  +ij.  +  i  is  even;   then  there  are  two  cases  to  be 
considered,  viz.  (a)  when  i  is  even,  and  (6)  when  j  is  odd: 

(a)  When  i  is  even  an  even  number  of  t,  ,  .  .  .  ,  i.  must  be  odd,  and 
the  number  of  odd  t\  multiplied  by  odd  jx  must  be  even.  Therefore  the 
number  of  odd  ix  multiplied  by  even  jx  must  be  even.  All  those  factors  p^ 
in  (13)  for  which  tx  is  even  involve  only  even  multiples  of  T,  and  those  for 
which  j\  is  even  and  tx  odd  involve  only  odd  multiples  of  T.  Since  there 
must  be  an  even  number  of  these  terms  involving  only  odd  multiples  of 
T,  their  product  involves  only  even  multiples  of  T. 


58  PERIODIC    ORBITS. 

(6)  When  i  is  odd  it  follows  from  (14)  that  an  odd  number  of  t\  ,  ...,«, 
are  odd,  and  from  the  hypothesis  that  iJt+  .  .  .  +ij,+i  is  even  it  follows 
that  an  odd  number  of  t\  jl  ,  .  .  .  ,  ijs  are  odd.  A  term  tx jx  can  be  odd 
only  if  both  *'x  and  j^  are  odd.  When  ,;x  is  odd  the  term  involves  only  even 
multiples  of  r  whether  raised  to  an  odd  or  even  power.  Since  the  whole 
number  of  odd  ix  is  odd,  and  an  odd  number  of  them  are  multiplied  by 
an  odd  jx ,  it  follows  that  there  is  an  even  number  of  terms  p£  ,  where  jx  is 
even  and  ix  is  odd.  Therefore  their  product  will  be  cosines  of  even  mul- 
tiples of  T.  That  is,  in  the  right  member  of  (13)  the  coefficients  of  even 
powers  of  e  involve  only  even  multiples  of  T. 

It  is  easily  proved  in  a  similar  way  that  the  coefficients  of  odd  powers 
of  e  in  (13)  are  odd  multiples  of  T. 

Upon  integrating  (13),  it  is  found  that  v  is  expansible  as  a  power  series 
in  e  of  the  form 

CO 

v  =  cr+2vtei,  (15) 

<=0 

where  vt  is  a  sum  of  sines  of  even  multiples  of  T  when  i  is  even,  and  of  odd 
multiples  of  T  when  i  is  odd.  The  coefficient  c  is  unity  because,  the  ellipse 
being  fixed  in  space,  v  increases  by  precisely  2ir  in  a  period. 

36.  Direct  Construction  of  the  Solution. — Upon  substituting  equation 
(9)  in  (7)  and  arranging  in  powers  of  e,  we  obtain 


S 

J=0 


(16) 


+  [-6p0p2+3p1(l-Pl-6p*)+2pS(3-5p*)K 

+  [-6p0(p3+3pf+3p0p2-2p1)+3p2(l-2p1)-5p?(8Pl-2-3pD]e4  .  .  .  ,  ] 

where  p''  is  the  second  derivative  of  p}  with  respect  to  T.  Upon  equating 
coefficients  of  like  powers  of  e,  the  differential  equations  which  define  the 
several  coefficients  become 

(a)  P:+PO=O, 

(6)  pf+Pl  =  l-3P*, 


Ps"+P3  =  - 


The  only  solution  of  (a)  satisfying  (10)  is 

P0  =  cosr.  (17) 

Then  equation  (6)  becomes 

pr+ft=-{-f  cos2r. 
The  solution  of  this  equation  satisfying  (10)  is 


KI.UPTIO  ~M<>TI<>\.  .V.) 

In  a  similar  MIMIUKT  «|ii;itions  (c),  (d),  .  .  .  can  be  integrated  in  order, 
and  their  solutions  are  found  to  In 


P3=     (-eos  T+cos3T),  p3=  -(co8  2T-cos  4r),         (19) 


The  general  term  of  die  solution  is  defined  by  an  equation  of  the  form 

C08T+    •    •    •    +  A?  CMltr,  (20) 


where  the  A(^  are  known  constants.  Since  p,  is  periodic,  A^  is  zero  for  all 
values  of  j,  and  since  p  involves  only  even  or  odd  multiples  of  T  according  as 
j  is  odd  or  even,  all  the  A(S  with  even  subscripts  are  zero  if  j  is  odd,  and  all  the 
A™  with  odd  subscripts  are  zero  if  /  is  even.  On  putting  A\"  equal  to  zero, 
the  solution  of  (20)  satisfying  the  initial  conditions  (10)  is 

pJ  =  A?+  \  -A?+  2  .A'"  1  COST-  2  xi1*    cosXT  (21) 

L  f^x~1J  £;  A  -1 

On  substituting  (17),  (18),  (19),  .  .  .  in  (9)  and  (5),  the  final  expression 
for  r  becomes 

r  =  a  \  1  -  [  cos  T]  e-f  ^  [  1  —  cos  2r  ]  e'+  -  [  cos  T  —  cos  3  T  ]  e* 

i  r  ~i 


|[cos2T-cos4r]  e*+  •  •  -\- 


(22. 


On  making  the  same  substitutions  in  (8)  and  integrating,  the  explicit  value 
of  v  is  found  to  he 


v  =  T+  [2  sin  T~\e+  |~4  sin  2rleJ4-  ["-  7  sin  T+  JJ  sin  3rlc3 

L  J  L  4  J  L         4  1  -  J 


(23) 


37.  Additional  Properties  of  the  Solution.  —  It  will  be  proved  that  no 
p,  carries  a  higher  multiple  of  T  than  j+l.  It  has  been  seen  that  it  is  true 
for  j  =  0,  1,  2,  3.  It  will  be  assumed  that  it  is  true  up  to  j  -  1,  and  then  it 
will  be  proved  that  it  is  true  for  the  next  step. 

The  general  term  in  the  right  member  of  (7)  is,  apart  from  its  numerical 
coefficient,  p'"le'.  After  substituting  the  series  (9)  for  p,  any  term  of  degree 
j  in  e  arising  from  this  term  has  the  form  Pop^pj1  •  •  •  p^e',  where 


X+X,+X,+  •  •  •  +X.  =  i*l,  X,+2XJ+  •  •  • 

After  eliminating  i  from  equations,  it  is  found  that 

X+2X,+3X,+  •  •  •  +(«+l)X.-j*l,  (24) 

where  obviously  a<j. 


60  PERIODIC    ORBITS. 

By  hypothesis  the  highest  multiples  of  r  in  p0  ,  p,  ,  .  .  .  ,  pK  are  respec- 
tively 1,2,  .  .  .  ,  K+l.  Therefore  the  highest  multiples  in  pi,  p*',  .  .  .  ,  pi'  are 
respectively  X,  2\lt  .  .  .  ,  (/c+l)XK.  Consequently  the  highest  multiple  in 
the  product  p^p*'  .  .  .  pi"  is  X+2X,+  •  •  •  +  (*+l)X10  which  is  j±\  by  (24). 
Therefore  the  highest  multiple  in  the  expression  for  pj  is  j  +  1. 

A  similar  discussion  of  (8),  (13),  and  (15)  shows  that  v}  does  not  involve 
multiples  of  T  greater  than  j. 

38.  Problem  of  the  Rotating  Ellipse.—  In  certain  cases  where  the  motion 
is  not  strictly  elliptical,  it  is  convenient  to  suppose  the  body  moves  in  an 
ellipse  whose  position  and  form  are  constantly  changing.  This  conception 
is  at  the  foundation  of  the  theory  of  perturbations  originated  by  Newton, 
and  has  been  essential  in  the  work  of  most  writers  on  celestial  mechanics. 

One  of  the  historically  interesting  and  important  problems  has  been  the 
theory  of  revolution  of  the  line  of  apsides  of  the  moon's  orbit.  When 
Clairaut  first  made  a  computation  of  the  rate  of  this  revolution  under  the 
supposition  that  it  was  due  to  perturbations  of  the  moon's  motion  by  the 
sun,  he  obtained  an  amount  about  half  as  great  as  that  furnished  by  obser- 
vations.* Later  work  by  himself  and  others  has  shown  that  the  discrepancy 
was  due  to  imperfections  in  his  theory,  but  at  first  he  sought  to  relieve  the 
difficulties  by  supposing  that  gravitation  does  not  vary  simply  as  the  inverse 
square  of  the  distance,  but  that  it  also  depends  upon  a  term  in  the  inverse 
third  power  of  the  distance.  We  shall  solve  the  problem  of  the  motion  for 
this  law  of  force  as  a  further  illustration  of  the  power  and  simplicity  of  the 
methods  which  are  being  used  here.  The  steps  in  the  solution  are  almost 
exactly  parallel  to  those  used  above,  and  it  will  be  noted  in  the  course  of  the 
work  that  they  would  not  be  fundamentally  different  if  the  added  term  were 
not  such  as  to  make  the  peculiar  simplicity  of  this  problem. 

The  differential  equations  of  motion  in  this  case  are 


,  t        2  ,_  /„« 

'       ~?-  ~^~     aM'          dt\Tt)~ 

where,  for  simplicity  in  the  final  formulas,  the  constant  coefficient  of  the  term 
in  the  inverse  third  power  of  r  is  given  the  form  k'(ml+ma)afji.  The  M  is  an 
arbitrary  parameter. 

The  integral  of  the  second  equation  of  (25)  is 

r'f-c,  (26) 

by  means  of  which  the  first  equation  reduces  to 

d2r      c2      F 


*See  Tiaserand's  Mecanique  Celeste,  vol.  Ill,  chap.  4,  and  particularly  articles  24  and  27. 


THE    ROTATING    ELLIPSE.  61 

39.  The  Circular  Solution.  We  shall  first  find  a  solution  of  (27)  with 
an  arbitrary  constant  of  areas,  c,  for  which  /•  is  mnMarit.  Let  a  and  u  be 
defined  by 

kt(ml+mt)=uta*l  c-kVmt  +  ni,)a.  (28) 

Let  r  =  a(l+p),  where  p  is  a  constant.     Then  (27)  becomes 

_L_        _!_  JL 

U+P)'      (l+p)'         (1+p)1' 

or,  expanding  as  a  power  series  in  p, 

p-3p2+6//-      •  •   =  -M(l-3p+         •)•  (29) 

By  §§1  and  2  this  equation  can  be  solved  for  p  as  a  power  series  in  n,  con- 
verging for  |/i|  sufficiently  small.  In  fact,  the  additional  term  has  been 
chosen  in  such  a  way  that  the  power  series  reduces  to  a  single  term,  but  it 
is  evident  that  this  condition  is  in  no  way  essential  to  the  process.  It  is 
found  at  once  that 

P=-M.  (30) 

In  this  case  the  solutions  of  equations  (25)  and  (26)  are 

(«-T),      (31) 


involving  the  three  constants  of  integration  a,  va  ,  and  T. 

40.  Existence  of  the  Non-Circular  Solutions.  —  We  shall  now  derive  a 
solution  of  equations  (25)  corresponding  to  the  elliptic  solution  in  the  ordi- 
nary two-body  problem.  It  will  involve  four  constants  of  integration  which 
are  arbitrary  except  for  the  restriction  that  the  orbit  shall  not  deviate  too 
widely  from  a  circle,  a  condition  which  is  imposed  to  secure  convergence  of 
the  series.  The  solution  is  thus  seen  to  be  the  general  solution. 

Now  let 

,  +m1)-wV, 


where  T  is  the  time  of  passing  the  nearest  apse,  and  a(l  —  p—  e)  is  the  arbi- 
trary initial  value  of  r.  Therefore  the  initial  value  of  p  is  unity.  The 
constants  a  and  e  are  defined  by  the  first  and  third  equations  at  t=  T,  u>  by 
the  second,  and  5  is  a  parameter  to  be  determined  later.*  There  are  so  far 
three  arbitrary  constants  of  integration  a,  e,  and  T;  the  fourth  is  introduced 
in  integrating  equation  (26). 

With  these  substitutions  equation  (27)  becomes 


p-  (33) 


•Poincan*  introduces  a  parameter  r  somewhat  analogous  to  ibis.     Let  Mfiltodet  NowtUe*  dt  la  Mfcanupu 
COetle,  vol.  I,  p.  61. 


62  PERIODIC    ORBITS. 

Equation  (33)  admits  a  periodic  solution,  as  is  known  from  the  fact  that  the 
orbit  is  a  rotating  ellipse.  However,  it  will  be  shown  directly  by  forming 
the  first  integral  of  (27),  viz., 

aM=^(r)-      (34) 


Suppose  the  initial  conditions  are  real;  then  <p(r0)>0.  For  /x  =  0  the  equa- 
tion <p(r)=Q  has  two  roots,  viz.,  r,  =  a(l  —  e)  ,  r2  =  a(l-\-e)  between  which  r 
must  vary.  For  fj.  small  it  also  has  two  roots,  r{  and  r'2,  near  i\  and  r2  respec- 
tively. Suppose  r{  <  r'2 .  Then  <p  (r)  <  0  if  r  <  r{  or  r  >  r'2 .  Consequently  it 
follows  from  dr/dt  =  V<p(r)  that  if  r  is  increasing  at  t  =  0  it  will  increase  until 
r  =  r'2  when  the  radical  changes  sign,  after  which  it  will  decrease  until  the 
radical  changes  sign  again  at  r  =  r{.  The  period  of  a  complete  oscillation  is 

(35) 

If  r  is  periodic  then  p  is  periodic  also. 

The  existence  of  the  periodic  solution  can  be  established  directly  from 
(33)  and  a  proof  made  of  the  possibility  of  a  construction  similar  to  that  used 
in  treating  the  elliptic  motion.  Equation  (33)  can  be  expanded  in  the  form 


,     ,    (36) 
-3p+3(l-4P2)e+-  •  -]M+[-6p+6e+ 


where  the  right  member  converges  so  long  as  |/*-fpe|<l.  By  §§14-16 
this  equation  can  be  integrated  so  as  to  express  p  as  a  power  series  in  8,  n, 
and  e  of  the  form 

P  =  P(«,  M,  e;  r).  (37) 

The  series  will  converge  for  all  T  in  the  interval  O^T^2?r  if  e\,  |/*  ,  |5j 
are  sufficiently  small. 

We  now  avail  ourselves  of  the  arbitrary  parameter  6  to  determine  the 
period.  We  will  determine  5  so  that  the  period  shall  be  2ir  in  T.  Since 
equation  (36)  does  not  involve  T  explicitly,  sufficient  conditions  for  perio- 
dicity with  the  period  2?r  are 

p(2ir)=P(0),  p'(27r)=p'(0).  (38) 

These  equations  are  not  independent,  for  (36)  has  an  integral  corresponding 
to  (34),  which  is  a  relation  between  p'  and  p  that  is  always  satisfied.* 
This  integral  has  the  form 

P/2  +  (l  +  S)p2  =  (l  +  S)P(p,  e,M)  +  C.  (39) 

'Compare  Les  Methodes  Nouvelks  de  la  Mtcanique  Celeste,  vol.  I,  p.  87. 


THE    ROTATING    ELLIPSE.  63 

Suppose  the  particle  is  projected  from  an  apse  so  that  p'  =  0  at  r  =  0.  Then 
C  is  a  power  series  in  c,  n,  and  the  initial  value  of  p.  which  is  unity.  Let  the 
iceneral  value  of  pbe  l+a.  Then  the  value  of  the  integral  at  any  time  minu- 
its  value  at  T  =  0  will  he  p'1  plus  a  power  series  in  a,  r,  and  »,  vanishing 
for  <r  =  0  whatever  c  and  n  may  be.  The  term*  coming  from  the  right 
member  all  involve  c  or  n  as  a  factor,  but  there  is  a  term  coming  from  p* 
which  involves  a  alone  to  the  first  degree.  Therefore  (39)  can  be  solved 
uniquely  for  a  as  a  power  series  in  pn  ,  e,  n,  vanishing  with  p'  =  0.  Hence 
if  p'  =  0at  T  =  2ir,  then  will  the  first  equation  of  (38)  necessarily  be  satisfied, 
and  consequently  it  may  be  suppressed. 

This  result  can  also  be  shown  from  a  certain  symmetry  which  is  par- 
ticularly simple  in  the  present  problem.  Equation  (36)  can  be  written 
in  the  form 


e,M).  (40) 

Suppose  p  =  1  ,  p'  =  0  at  r  =  0  and  that  these  equations  have  the  solution 
P-/,(T),P'-/,(T). 

Now  consider  the  differential  equations  obtained  when  (40)  are  trans- 
formed by  the  substitution  p  =  p,  ,  p'--p(,  T=-T,.  The  equations  in  the 
new  variables  are  the  same  as  in  the  old,  and  consequently  if  the  initial 
conditions  are  the  same  (p,  =  1,  p(  =  0  at  r  =  0),  the  solution  is 


Therefore  p  is  an  even  function  of  r  and  p'  is  an  odd  function  of  T. 

Suppose  p'  =  0  at  T  =  r.  Since  it  is  an  odd  function  it  must  also  have 
been  zero  at  r  =  —  r.  Since  p  is  even  in  T  it  has  the  same  value  at  T  =  —  T  as 
it  has  at  T  =  T.  Consequently  the  system  is  the  same  at  T  =  T  as  it  was  at 
r=—  T,  and  the  motion  is  periodic  with  the  period  2*.  Hence  if  p'  =  0 
at  T  =  0,  it  is  sufficient  to  satisfy  the  condition  p'  =  0  at  T  =  T  in  order  to 
secure  a  periodic  solution  of  (33)  with  the  period  2ir. 

It  will  now  be  shown  that  the  second  equation  of  (38)  can  be  solved 
uniquely  for  8  as  a  power  series  in  n  and  e,  vanishing  with  M  =  0.  It  is  found 
by  integrating  (36)  and  imposing  the  initial  conditions  p  ^  1,  p  f;  0  that 


where 

Poo  =  COS  V/l  +  6 

Pio=  -  o 


(41) 


p'(2w)  -  p'  (0)  =  d  [-  I  TT  +  (terms  in  6,  M, 

—  3  TT  +  (terms  in  d,  n,  e)]  =  0. 


64  PERIODIC    ORBITS. 

On  substituting  these  results  in  the  second  equation  of  (38)  and  expanding 
in  powers  of  5  also,  we  get 


(42) 


There  are  no  terms  in  e  alone,  for  when  M  =  0>  the  orbit  becomes  a  fixed 
ellipse  and  5  =  0  satisfies  the  periodicity  condition.  Hence  (42)  can  be 
solved  uniquely  for  8  as  a  power  series  in  n  and  e  ,  vanishing  with  M  =  0  . 
When  the  value  of  5  obtained  from  (42)  is  substituted  in  (37),  p  becomes 
periodic  in  T  with  the  period  2ir ,  and  is  expanded  as  a  power  series  in  n  and  e 
which  converges  provided  |M)  and  \e\  are  sufficiently  small.  It  can  be  written 

P=Z    I  P(,MV,  5=  S    2  50.MV.  (43) 

t=0    j=0  1=1    1=0 

From  the  reasoning  of  §35  it  follows  that  each  p(J  separately  is  periodic. 

The  range  of  convergence  of  (43)  is  limited  in  two  ways.  In  the  first 
place  the  inequalities  1 6 1  <  50 ,  fj.\<fi0  ,  \e  <e0  must  be  satisfied  in  order  that 
(37)  may  converge  for  0  =  r^27r.  Then  the  inequalities  |M|<MI  >  lel<ei 
must  be  satisfied  in  order  that  the  solution  of  (42)  shall  converge  and  give 
for  1 5 1  a  value  less  than  50 .  When  |  /z  and  |  e  \  satisfy  both  of  these  sets  of 
inequalities  the  convergence  of  (43)  is  assured  for  all  T. 

After  the  explicit  development  of  equations  (43)  has  been  made  the 
results  can  be  substituted  in  (26),  when  v  will  be  determined  by  a  quadra- 
ture. The  final  form  of  v  is 

v=  2  1  Vij^ei  (44) 

i=0  j=0 

The  constant  parts  of  the  vt)  are  independent  of  e  since  f or  n  =  0  it  was 
found  that  v  =  T  +  periodic  terms. 

41.  Direct  Construction  of  the  Non-Circular  Solution.— In  carrying  out 
the  practical  construction  of  the  solution,  we  shall  make  use  of  the  facts  that 
(a)  p^l,  p'^0  at  r  =  0,  (6)  p  is  expansible  in  the  form  (43),  and  (c)  each 
ptj  separately  is  periodic  with  the  period  2ir. 

On  substituting  (43)  in  (36)  and  equating  coefficients  of  equal  powers 
of  /JL  and  e,  it  is  found  that  the  several  coefficients  must  satisfy 

(A) 
(B) 


(D)  Po2 

— 3p01+3(l  — 


"00) 


llli:    UOTATINc;    KI.Ul'SE.  <).") 

The  solution  of  1  -atisfying  (a)  is  pw  =  cos  r.  The  solution  of  (B)  is 
given  in  equation  (18).  The  solution  of  ((')  is  not  periodic  unless  the  coeffi- 
cient of  POO  is  zero.  Imposing  also  the  condition  (a),  i,0  =-  —  3,  p,0  =  0. 
The  term  pM  is  given  in  equation  (1!)).  Equation  (E)  becomes  explicitly 
PM  +  PII  =  ~3  <''*~  S,,COST  —  3/2  cos2r.  Upon  imposing  conditions  (a)  and 
(c),  the  solution  of  this  equation  is  found  to  be 

5,,  =  0,  PlI=-f+co8T+}co82r.  (45) 

The  explicit  form  of  (/•')  now  becomes  P»+P»  =  (—  $jo+3)  coe  T,  whose 
solution  satisfying  the  conditions  (a)  and  (c)  is 

*»  =  3,  p»  =  0.  (47) 

Hence  the  final  expressions  for  p  and  6  as  power  series  in  c  and  M  are 

p  =  cosT-f  [-|  +  {  cos27-]e+ [- f +  C08T  +  I  co82r]ne 

+  f[-cosT+cos3T]e'+  •  •  •  , 


where,  from  (32),  T  is  to  be  replaced  throughout  by  u(t-  r)/\/l-j-j. 
The  differential  equation  defining  the  general  term  is 


Suppose  all  the  p.*  and  6^  for  which  <c<»,  X<^'  have  been  found.      Then 
this  equation  can  be  written  in  the  form 

there  being  no  sine  terms.     Its  solution  satisfying  conditions  (a)  and  (c)  is 

A      -  A VJ)  1 

1  (50) 

The  5lt  and  all  the  coefficients  are  uniquely  determined.     Therefore  the 
process  can  be  continued  without  modification  as  far  as  may  be  desired. 
After  the  transformations  (32)  equation  (26)  becomes 

dv 
dr 

Upon  substituting  the  value  of  p  given  in  (48)  and  integrating,  it  is  found 
that 


t,_t,o=[l+2ju+3M'+  '  '  •]T+[28inT]e+[fsin2T]e'+  •  •  •  •     (51) 

where  T  is  to  be  replaced  by  u(t-  T)/Vl  +  S.    The  four  arbitrary  constants 
of  integration  are  a,  e,  T,  and  v0 . 


66  PERIODIC    ORBITS. 

42.  Properties  of  the  Solution. — It  has  been  proved  that  each  ptj 
(except  POO)  and  p'fj  vanish  at  r=Q,  that  each  pis  is  periodic  in  T  with  the 
period  2ir,  and  that  p  is  an  even  function  in  T.  Hence  each  p,;  involves  only 
cosines  of  multiples  of  T.  It  has  also  been  noted  that  d  depends  upon  no 
terms  independent  of  M- 

There  is  no  term  p,0  distinct  from  zero,  for  when  e  =  0  the  differential 
equations  are  satisfied  for  the  same  initial  values  of  the  variables  by  p  =  0. 
Or,  it  is  seen,  from  the  development  of  the  second  term  of  (33),  that  the  only 
terms  which  are  independent  of  e  involve  p  to  the  first  degree  alone.  Conse- 
quently the  right  members  of  the  equations  corresponding  to  (49),  which 
define  p(0,  will  be  [  —  &m+f(p)]  cos  T  alone.  The  periodicity  condition  makes 
it  necessary  to  put  Slo=f(n~) ,  and  the  initial  conditions  then  make  p,0  =  0. 

The  expression  for  the  right  member  of  (51)  has  no  term  independent 
of  M,  except  unity,  in  the  non-periodic  part,  for  when  /x  =  0  the  ellipse  is 
fixed  and  this  part  reduces  simply  to  T.  On  making  use  of  all  of  these  facts 
and  some  simple  artifices,  the  labor  of  actually  constructing  the  series  can 
be  very  much  reduced. 

The  radius  r  completes  its  period  in  T  =  2  TT,  or  t  =  2irVl  -f  5/w.  It  follows 
from  (51)  that  the  longitude  of  the  radius  has  increased  in  this  interval  by 
vT+5  [l+2/i+3/i2+  •  •  -]27r.  Therefore  the  line  of  apsides  has  moved 
forward  in  this  interval  through  the  angleVi+5  [l+2/*+3/i2+  •  •  •  ]  2ir  —  2ir. 
Hence  its  average  rate  of  angular  motion  in  t  is 

<   ]    27T-27T 

dt  ~ 


where  w  is  the  longitude  of  the  nearest  apse.     The  parameter  fj,  can  be 
determined  so  as  to  secure  any  rate  of  revolution  of  the  apsides  not  too  great. 


CHAPTER  III. 

THE  SPHERICAL  PENDULUM. 
I.    SOLUTION  OF  THE  Z-EQUATION. 

43.  The  Differential  Equations.  The  problem  of  I  lie  spherical  pendu- 
lum falls  in  the  class  of  those  which  can  be  treated  by  the  methods  of 
periodic  orbits.  It-  simplicity  makes  it  particularly  well  suited  to  illustrating 
the.-e  proeesse.-,  and  its  value  as  an  introduction  to  the  subject  is  increased 
by  the  fact  that  it  is  easy  to  verify  the  results  experimentally.  It  is  doubt- 
ful whether  there  is  a  problem  which  is  superior  in  these  resjxxjts. 

Let  us  take  a  rectangular  system  of  axes  with  the  jx^sitive  2-a\i- 
directed  upward  and  with  the  origin  at  the  fixed  point  of  the  pendulum. 
The  pendulum  is  subject  to  gravity  and  the  normal  reaction,  N,  which 
we  shall  take  with  the  positive  sign  when  directed  outward.  If  we  represent 
the  radius  of  gyration  by  /,  the  motion  of  the  pendulum  satisfies 

j+jf+f^?,          mx'  =  N^,        my"  =  N^,        im'  =  Nl-mg,     (1) 

where  the  accents  indicate  derivatives  with  respect  to  the  time. 
The  last  three  equations  of  (1)  admit  the  integral 

,  (2) 


where  c,  is  the  constant  of  integration. 

The  normal  reaction  exactly  balances  the  centrifugal  acceleration  of  the 
pendulum  due  to  its  motion  and  the  component  of  mg  along  the  normal  to 
the  surface;  hence 

mv1  Hi  r'          z 


*»  I  JV  1 

where  Fff  is  the  normal  component  of  mg.     On  making  use  of  (2),  we  get 

\T    ™Q  (i,    ~  ^  f<N 

^V=  —^-(3z— c.)~  W 

I 

Hence  equations  (1)  become 


x'  = 


(4) 


The  last  equation  is  independent  of  the  others  and  is  therefore  solved  first. 
After  it  is  solved  the  second  gives  x  in  terms  of  I,  and  then  y  can  be  found 
from  the  first.  » 


68  PERIODIC    ORBITS. 

44.  Transformation  of  the  z-Equation. — It  will  be  convenient  to  trans- 
form the  last  equation  of  (4).     It  admits  the  integral 

2'2=|(2z-Cl)z2-</(2z-c2)=/(z),  (5) 

where  c2  is  a  constant  of  integration  which  is  independent  of  c^     If  we 
subtract  this  equation  from  (2)  and  reduce  the  result  by  (2),  we  find 

-2  ~2 

-jz'2.  (6) 

Now  z2  ^  I2.  In  the  case  where  the  spherical  pendulum  reduces  to  the  simple 
pendulum  z  takes  the  value  =  I,  and  at  the  same  time  z'  =  0 .  In  this  case 
Cj  —  c2  =  0.  In  the  spherical  pendulum  z>  —  I  when  z'  =  0;  consequently  in 
this  case  ct  — c2>0.  Hence  in  all  cases  of  the  physical  problem  c,  —  c2^0. 
Now  consider  equation  (5).  If  the  initial  conditions  are  real,2o2=/(z0) 
is  zero  or  positive  and 


>)  =  +00. 

Therefore  the  equation  /(z)  =0  has  always  three  real  roots.  Let  them 
be  ttj  ,  o2  ,  and  a3  ,  where  the  notation  is  chosen  so  that  e^  ^  a2  ^  o3  .  Then 
equation  (5)  becomes 


On  comparing  this  equation  with  (5),  it  is  seen  that 

2(al  +  o2+a3)=c1,          alaa+aaa3+a3al  =  -f,          2  at  a2a  3=  -C2  f  .      (7) 

It  will  now  be  shown  that  a3  satisfies  the  inequalities  —  /  ^a3^  0.  It 
follows  from  the  last  of  (7)  that  c2  is  negative  if  a3  is  positive.  On  putting 
z'  =  Q  and  z  =  a3  in  (5),  we  get 


By  hypothesis  the  first  term  on  the  right  is  positive,  and  by  (2)  the  second 
can  not  be  negative.  Therefore  c2  must  be  positive,  which  contradicts  the 
implication  from  the  last  of  (7).  Therefore  a3^0. 

Some  special  cases  may  be  indicated  : 

(1)  It  follows  from  (5),  (6),  and  (7)  that  a3  =  -/  implies  that  at=  +1, 
2a2  =  ci  =  c2,  at  —  a3  =  2L  The  constant  a2  is  not  determined,  and  we  shall 
suppose  it  is  less  than  +1.  This  case  is  that  of  the  ordinary  simple  pendu- 
lum making  finite  oscillations.  In  the  sub-case  where  a2  =  —  I,  we  have 
c,  =  ca  =  —  2f  and  the  solutions  of  (4)  are  x  =  y  =  Q,  z=—  I. 


THK    M'llKRICAL   PENDULUM.  fi«l 

(2)  Ifdj      (/,  it  follows  from  the  same  (Munition-  that  cij-     /, 'Jc^-C^C,. 
The  constant  a,  is  not  determined.      If  at>l,  we  have  the  case  of  the  simple 
|)eii<luluin  swinging  round  and  round,  and  a,-as>2/.    In  the  sub-case  where 
a,  =  /,  we  have  c,=c,=  +2/,  and  the  solutions  of  (4)  are  x-y-0,  z—+l. 

(3)  If  a,  =  0,  it  follows  that  o,^0.     Therefore  the  second  of  (7)  can 
not  be  satisfied  except  by  oa  =  0,  a,  =  oo .    Then,  from  the  first  equation  we 
net  <-,  =  <».     This  is  the  case  of  revolution  in  the  xt/-plane  with   infinite 
>l>erd,  and  of  course  can  not  be  n-ali/ed  physically.     Excluding  this  case 
and  that  of  the  simple  pendulum,   the  constants  o, ,  a,,  as  satisfy  the 
inequalities  — /<o,<0,    — /<a,<+/,   a,>-f/- 

Now  make  the  transformation 

2-o,=  (o1-o,)ttt.  (8) 

Then  equation  (5)  becomes 

(9) 


fll  — 

Also  let 

_  ot-o.  yd 

p  *-w 

where  t0  is  an  arbitrary  initial  time  and  5  is  a  constant  as  yet  undefined. 
The  constant  n  satisfies  the  inequalities  O^M^!-    Then  (9)  becomes 

u*=  (!+«)(! -u*)  (l-Muf)=F(tt),  (11) 

where  it  is  the  first  derivative  of  u  with  respect  to  the  new  independent 
variable  r.    The  first  derivative  of  (11)  is 

'].  (12) 


45.  First  Demonstration  that  the  Solution  of  (12)  is  Periodic,  and  that 
u  and  the  Period  are  Expansible  as  Power  Series  in  p. — It  will  first  be 
shown  that,  for  any  initial  conditions  belonging  to  the  physical  problem, 
except  when  /*  =  !,  the  solution  of  (12)  is  periodic.  By  the  fundamental 
existence  theorem  of  the  solutions  of  differential  equations*  the  solutions  of 
(12)  are  regular  in  T  for  all  finite  values  of  r  and  u.  For  real  initial  condi- 
tions the  coefficients  of  u  and  its  derivatives  expanded  as  power  series  in  r 
are  real,  and  by  analytic  continuation  they  remain  real  for  all  finite  real 
values  of  T  provided  u  does  not  become  infinite.  Now  consider  the  curve 
F  =  F(u).  Suppose  u  =  u0  at  r  =  Q  and  that  «0  is  positive.  Then  u  is 
increasing  at  a  rate  which  is  proportional  to  the  square  root  of  F(u0),  and  it 
continues  to  increase  until  u=l.  It  can  not  increase  beyond  +1  for  then  u 
would  become  a  pure  imaginary,  and  it  has  just  been  shown  that  it  always 
remains  real.  It  can  not  remain  constantly  equal  to  1  unless  M=l;  for 
otherwise  u  =  1  does  not  satisfy  (12).  Therefore,  unless  n=l,u  will  increase 

•Picard's  Trait*  fAnalyte,  vol.  II,  chap.  11,  fill. 


70  PERIODIC    ORBITS. 

to  1  and  then  decrease  to  —  1 ;  then,  in  a  similar  way,  it  changes  at  u  =  —  1 
from  a  decreasing  to  an  increasing  function.  That  is,  at  u=  ±1  the  func- 
tion F(u)  changes  sign  and  u  varies  periodically  between  +1  and  —  1. 

This  result  follows,  of  course,  from  the  fact  that  in  the  present  problem 
u  is  the  sine  amplitude  of  T,  one  of  whose  properties  is  that  of  having 
a  real  period,  but  the  argument  given  above  applies  to  much  more  general 
cases,  and  the  result  can  be  read  from  the  diagram  for  F  =  F(u).  It  may 
be  mentioned  in  passing  that  the  imaginary  period  of  the  elliptic  function  is 
associated  in  a  similar  way  with  the  portions  of  the  curve  between  + 1  and 
+  1/Vju,  and  between  —  1/V/x  and  —  1 . 

The  period  of  a  complete  oscillation  is  found  from  (11)  to  be 

P=    _2_    f+1  du 

../i r~s  I      -.//i     ..2\/i       _.2\'  \iOJ 


which  is  finite  unless  /z  =  1  .     We  shall  exclude  this  exceptional  case.     It  is 
well  known  that 


D 

= 


—  1) 


i 
T 


In  t  the  period  is 

i-sV  „,  1 

)  M  +  •  •  -J-     (is) 


That  is,  the  period  is  expansible  as  a  power  series  in  ju,  and  in  the  present 
simple  case  the  series  converges  provided   n\<l. 

The  constant  5  has  so  far  remained  undetermined.     If  we  let 


the  period  in  r  will  be  simply  2ir.     On  solving  this  equation,  we  find  that 
the  required  value  of  5  is 


which  is  a  power  series  in  /i. 

Now  consider  equation  (12).  By  §§14-16,  this  equation  can  be  inte- 
grated as  a  power  series  in  n,  and  |  /x  |  can  be  taken  so  small  that  the  series 
will  converge  for  all  T  in  the  interval  0  ^  T  5S  TI  chosen  arbitrarily  in  advance. 
If  r,  is  greater  than  P  and  the  constant  of  integration  is  chosen  so  that 
(11)  is  the  first  integral  of  (12),  that  is,  so  that  the  period  of  the  motion  is 
P,  then  it  follows  from  the  periodicity  of  the  solution  that  the  series  will 
converge  for  all  finite  values  of  r.  That  is,  both  u  and  T  are  expansible 
as  power  series  in  p. 


THE    SPHERICAL   PKMHI.UM.  71 

46.  Second  Demonstration  that  the  Solution  of  Equation  (12)  is  Peri- 
odic. -While  the  proof  of  j  I.")  N  siillicient  for  the  const  ruction  of  the  solution, 
it  will  he  instructive  to  give  another  demonstration  of  its  periodicity.  In 
many  problems  the  former  methods  can  not  he  applied. 

By  §§14-l(i,  equation  (12)  can  he  integrated  as  a  power  series  in  the  two 
parameters  /j.  and  o  of  the  form 

n=SoZu4/(T)«V,  (17) 

where  the  utl  are  functions  of  T  to  be  determined  by  the  conditions  that 
(17)  shall  satisfy  (I'J)  and  the  initial  conditions,  and  where  \6\  and  |M|  can 
be  taken  so  small  that  the  series  will  converge  for  any  interval  O^T^T,, 
chosen  arbitrarily  in  advance.  We  may  take  the  initial  values  u(0)  =  0, 
u(0)=a,  and  from  (11)  we  see  that  in  order  to  get  the  same  solution  as 
before  we  must  put  a=  Vl-f  &  at  the  end.  The  subsequent  steps  of  this 
demonstration  would  not  be  essentially  modified  if  we  took  general  initial 
conditions.  From  these  initial  conditions  we  get 

0=S    2u,,(0)«V,  fl-2   2«,,(0)«V, 

(-0    J-0  (-0    >-0 

from  which  it  follows  that 

u(>(0)  =  0      (t,j-0,  •  •  •  oo),        Woo(0)=a,     u,,(0)=0      (i+j>0).  (18) 


On  substituting  (17)  in  (12)  and  equating  coefficients  of  corresjwnding 
powers  of  5  and  M,  we  get 


=  -"«>,  (19) 

The  solution  of  the  first  of  these  equations  satisfying  (18)  is 

MM  =  a  sin  T.  (20) 

Upon  substituting  this  result  in  the  right  member  of  the  second  equation 
of  (19),  integrating,  and  imposing  the  conditions  (18),  we  find 

w10=-  |sinT+  IT-COST.  (21) 

Hence  we  have 

u  =  asinT+  %  [_8jnT+  T  cos  T]  5  +  higher  powers  of  6  and  ».        (22) 

m 

Since  the  right  member  of  (12)  does  not  contain  T  explicitly,  sufficient 
conditions  that  u  shall  be  periodic  with  the  period  2  T  in  T  are 

«(2T)-u(0)=0,  M(2r)-M(0)=0.  (23) 


72  PERIODIC    ORBITS. 

It  will  now  be  shown  that  the  second  one  of  these  equations  is  necessarily 
satisfied  when  the  first  is  fulfilled.  Let  u  =  Q+v,  u  =  a-\-v,  where  v  and  v 
vanish  at  r  =  0.  Then  (11)  becomes 


On  making  use  of  the  fact  that  1  +  d  =  a,  we  have 


There  are  two  solutions  of  this  equation  for  v,  but  the  one  which  vanishes 
at  r  =  0  must  be  used.  It  has  the  form 

i>=vp(v),  (24) 

where  p(v)  is  a  power  series  in  v.  Since  u  (0)  =  v  (0)  =  0,  it  follows  from 
the  first  equation  of  (23)  that  w(27r)=0.  Then,  from  (24)  we  have 
v  (2ir)  =u(2ii)—  M(0)=0.  That  is,  by  virtue  of  the  existence  of  the 
integral  (11),  the  second  equation  of  (23)  is  a  consequence  of  the  first. 

Now  let  us  consider  the  solution  of  the  first  equation  of  (23).     Upon 
substituting  u  from  (22),  we  get 

0  =  Trad  +  terms  of  higher  degree  in  5  and  //.  (25) 

It  follows  from  the  theorems  of  §§  1-3  that  this  equation  can  be  solved  for  5 
uniquely  in  the  form 

(26) 


where  P,(M)  is  a  power  series  in  n,  which  converges  if  |M|  is  sufficiently  small. 
On  substituting  this  result  in  (17),  we  have 


J=0 


(27) 


which  converges  for  all  0<r^2ir  for  \n\  sufficiently  small.  It  is  sufficient 
that  |  M  and  1 5  satisfy  the  conditions  necessary  to  insure  the  convergence  of 
(17)  and  the  solution  of  (25).  These  conditions  can  both  be  satisfied  by 
values  of  /j.  different  from  zero  because  the  expression  for  5,  given  in  equa- 
tion (26) ,  carries  ^  as  a  factor. 

Hence,  the  periodicity  conditions  having  been  satisfied,  we  have  proved 
that  the  solution  is  periodic.  It  has  been  found  to  be  expansible  as  a  power 
series  in  n,  and  the  period,  which  in  t  is 


T= 

Vg(ai  —  as) 

is  also  expansible  uniquely  as  a  power  series  in  /*.  It  is  clear  that  this  mode 
of  demonstration  applies  to  a  wide  class  of  equations,  for  the  explicit  values 
of  only  the  first  terms  of  the  right  member  of  the  differential  equation,  the 
general  properties  of  its  convergence,  and  the  existence  of  a  first  integral 
have  been  used. 


I1IK    sl'HKHH-AL    PENDULUM.  73 

47.  Third  Proof  that  the  Solution  of  Equation  1  12)  is  Periodic.     Tin  -re 

is  a  certain  symmetry  property  of  the  solutions  \\hich  can  be  used  to 
-implify  (lie  demonstration  that  the  motion  is  periodic.  1(  will  be  shown 
that  the  motion  i>  >ymnic(ric:il  in  r  with  ropect  to  the  value  «  =  0,  which, 
by  (8),  corresponds  to  z  =  a3,  or  the  lowest  point  reached  by  the  pendulum. 
Suppose  //  =  (),  it=n  at  T  =  0  and  that  the  solution  of  (12)  for  these 
initial  conditions  is 


«  =  /,«,        /,(0)=0,        M-/f(r),        /,(0)-o.  (28) 

Now  make  the  transformation 

u=  —  v,        u  —  i),        T=—<r.  (29) 

Then  (12)  becomes 

i/  =  (l  +  «)[-a+M)H-2Mta  (30) 

Hence  if  r  =  0,  v  =a  at  <r  =  0,  the  solution  of  (30)  is 

a,  (31) 


where  /,  and  /2  are  the  same  functions  of  <r  that  /,  and  /,,  of  (28),  are  of  T. 
On  substituting  (28)  and  (31)  in  (29),  we  get 


Therefore  M  is  an  odd  function  of  T  when  u=0  at  r=0,  and  hence  it 
follows  that  sufficient  conditions  that  the  solution  of  (12)  shall  be  periodic 
with  the  period  2r  are 

w(0)=0,  u(x)=0.  (32) 

The  solution  (22)  was  obtained  with  the  initial  condition  w(0)=0.  Hence 
the  second  equation  of  (32)  becomes 

0  =  -  '  a5+  higher  powers  of  6  and  p.  (33) 

The  solution  of  this  equation  for  6  and  the  further  discussion  are  precisely 
like  the  treatment  of  (25),  and  lead  to  the  same  results. 

48.  Direct  Construction  of  the  Solution.-  It  has  been  proved  (hat  & 
can  be  expanded  as  a  j)ower  series  of  the  form 


'  '  '  (34) 

such  that,  when  (12)  is  integrated  as  a  power  series  in  M  of  the  form 


74  PERIODIC    ORBITS. 

with  the  initial  conditions  u  (0)  =  0,  u  (0)  =  a,  u  will  be  periodic  with  the 
period  2n.  In  fact,  the  value  of  5  is  given  in  (16),  but  we  shall  make  use 
only  of  its  expansibility  in  this  form;  in  more  complicated  problems  its 
explicit  value  would  not,  in  general,  be  known.  Since  u  is  periodic  with  the 
period  2ir,  we  have 

u(r}  =  S  [tt,(2ir  +  T)  -«, 


J=0 

and  this  equation  holds  for  all  |ju|  sufficiently  small.     Therefore 

Uj(2ir-\-T)  —  us(r)  =  0,  0  =  0,  1,  ...  oo).  (36) 

Hence  each  Uj  separately  is  periodic  with  the  period  2ir, 

Instead  of  determining  the  solution  by  the  initial  conditions  w(0)=0, 
u  (0)  =  0,  we  may  use  u  (0)  =  0,  u(r/2)  =  1.  Or,  by  (35) , 

oo 

S 

1=0 

Therefore  we  have 

«o(0)=0,         u0  (*)  =  !, 

(37) 
j=l,  2,  ...  oo). 

Now  we  determine  u  and  d  by  the  conditions  that  (12)  shall  be  satisfied 
identically  in  /x  and  T,  and  that  the  conditions  (36)  and  (37)  shall  be  ful- 
filled. By  direct  substitution  and  equating  of  coefficients,  we  find 


(38) 


u0-  (1  +  5,) 


The  solution  of  the  first  equation  of  (38)  satisfying  (37)  is  u0  =  sin  T. 
Then  the  second  equation  of  (38)  becomes 

-f  |  sinr-i  sin3r.  (39) 


2  2 


In  order  that  the  solution  of  this  equation  shall  be  periodic  we  must  set  the 
coefficient  of  sinr  equal  to  zero.     Then  the  solution  satisfying  (37)  is 

^i=2">  ui=  ~^f  [sinr+  sin3r].  (40) 

Upon  substituting  the  results  already  obtained  in  the  third  equation 
of  (38),  we  find 

sinr+  —  sinr—  ^  sin3r—  j-  sin5r.  (41) 


IMF.    SPHERICAL    PKNIH  I.UM.  75 

Upon  setting  the  coefficient  of  sinr  equal  to  xero.  as  In-fore,  and  integrating 
subject  to  the  condition-    :>7),  \ve  find 

5>  =  "w  '         "-  =  -i-  (7  sin  r  +  8  sin  3r  +  sin  ST].  (42) 

The  induction  to  the  general  term  can  now  he  made.  We  n>.-ume  that 
u0  ,  .  .  .  ,  «,_,  ;  5,  ,  .  .  .  ,  6,_,  have  been  determined  and  that  it  has  been 
found  that  i/  is  a  sum  of  sines  of  odd  multiplies  of  r,  of  which  the  highest 
is  2./  +  1.  The  differential  equation  for  the  coefficient  of  p  \e 

W,+M,=  -S,w0+F,(«,,  u*)  (k,  X-0  .....  i-l),      (43) 

where  the  Ft  are  linear  in  the  5,  and  of  the  third  degree  in  the  ux.     The 
general  term  of  /•',  is 


(44) 


where 

m  =  0  or  1, 

»i  +»•+*.=  lor  3, 
THK+J',  X,  +  i'1X,+i'IX>=  i  or  t-1  (vi+Vt+Vt**!  or  3). 


It  follows  from  the  second  of  these  equations  that  there  are  an  odd  number 
of  odd  v,  .  Consequently  T,  is  a  sum  of  sines  of  odd  multiplies  of  r.  The 
highest  multiple  of  r  is 


On  reducing  this  expression  by  the  third  of  equation  (44),  it  is  found  that 
N,=  —  2mK+2i  +  l,  the  greatest  value  of  which  is,  by  (44), 

JV,-2t+l.  (45) 

Hence  (43)  may  be  written 

l  sin(2i+l)r,        (46) 


where  the  A$+l  are  known  constants. 

In  order  that  the  solution  of  (46)  shall  be  periodic,  the  condition 

«,-X«  (47) 

must  be  satisfied,  which  uniquely  determines  5,.     Then  the  solution  of 
equation  (46)  satisfying  the  conditions  (37)  is 

sin(2i+l)r, 

l«> 

hi (\  =  \ >}. 

(48) 


4T(x+T) 


i(0=  7P  C-l)x 
»i       &  \ 


4X(X+l 


The  solution  at  this  step  has  the  same  form  as  that  which  was  assumed  for 
«„ ,  .  .  .  ,  u,_, ,  and  the  induction  is  therefore  complete. 


76  PERIODIC    ORBITS. 

On  collecting  results,  we  have  for  the  first  terms  of  the  solution 
•u  =  [sinr]+      [si 


(49) 


And  substituting  these  results  in  (8),  we  get  for  the  final  result 
\  Oi-o3)  [1  ~  C0s2r]  ft  +  -5  (o,-  as)  [1  -cos4r]  £ 

0i-a3)  [16+3  cos2r-16cos4T-3cos6r]M3+  •  •  •  ; 


(50) 


the  expression  for  71  agreeing  with  that  found  in  (15). 


49.  Construction  of  the  Solution  from  the  Integral.  —  In  the  direct 
construction  of  the  solution  we  have  made  no  explicit  use  of  the  existence  of 
the  integral  (11).  We  shall  show  that  it  can  be  used  to  check  the  computa- 
tions, or  to  furnish  the  solution  itself. 

Equation  (11)  can  be  written  in  the  form 


Since  u,  u,  and  8  can  be  expanded  as  converging  power  series  in  n,  we  have 

V  =  <Po+<PilJi+<f>2H2+  •  •  •   =  0-  (51) 

Since  this  equation  is  an  identity  in  /x,  it  follows  that 

<f>i  =  ft(u0  ,u0,d2,  .  .  .  ,u(,u(,dt)  =  Q         (i=o,  ...«),        (52) 

where  <pt  is  a  polynomial  in  u0  ,  .  .  .  ,  dt  .  It  follows  from  (11)  that  <pf 
is  linear  in  the  5X,  of  the  second  degree  in  MX,  and  MX,,  and  of  the  second  or 
fourth  degree  in  MX,  and  MX,.  Therefore  <pt  is  a  sum  of  cosines  of  even 
multiples  of  r.  It  is  seen  without  difficulty  that  the  highest  multiple  of 
T  in  &  is  2t+2.  Hence  we  have 


=  0.  (53) 

Since  this  equation  holds  for  all  values  of  T,  it  follows  that 

££=0          (»•  =  !,  .  .  .  oo;  X=0,  .  .  .  ,7!+!).  (54) 


THK    SIMIKKH  AL   PENDULUM.  77 

The  fljxiin-  functions  of  the  o£+I  and  5,,  and  equations  (54)  constitute 
a  searching  check  upon  the  computation  of  these  quantities.  If  by  some 
numerical  accident  an  error  were  not  indicated  at  any  particular  step,  it 
would  he  revealed  by  the  failure  to  satisfy  (54)  at  some  later  step. 

Hut  the  a£x+,  can  be  computed  from (54), as  will  now  be  shown.  Suppose 
«„,...,  «,_,  ;  6, ,  .  .  .  ,  $„_,  have  been  computed.  Then  we  find  from 

the  explicit   expression  (11)  that 

V,  =  2iV0u1+2u0u(-«,(l-u;)-f^("x ,  Wx ,  **)          (A-0 »-l), 

the  \f/t  being  known  functions.     Hence,  using  the  notation  of  the  first  equa- 
tion of  (48),  we  get 


(X-2 


(55) 


where  the  C(£  (\  =  Q,  .  .  .  ,  i+1)  are  known  constants.      On  solving  these 
equations  beginning  with  the  last,  we  find 


en 

- 


•*-,-  -x±i«*-gs^D 

i  c-ioi  f)f(n          _<i> 1  i        1  /~<(n 

ot  —  ottj  -(-^Jl/j  ,         a,   --  —  o, —  'o^o  > 


(56) 


which  uniquely  determine  6,  and  the 

Let  us  apply  these  equations  to  the  computation  of  the  first  terms  of  u. 
Suppose  u0  =  sinT  and  take  t  =  l.     Then  we  find  from  (11)  that 


whence 


Therefore 


<n      l  *    I    l  _n         K<|>_4.<'>      1  *  _n         H(|>_  o-'"      *  —  n 
,   ••^*i+g  =  ">        «»   =-4a,  -•^6l  =  0,        a*  =Zat  —  -  *  U. 


agreeing  with  the  results  already  found.     The  process  can  be  continued  as 
far  as  may  be  desired. 

Two  different  methods  of  computing  the  solutions  have  been  developed. 
They  are  both  reduced  by  the  general  discussion  to  the  mere  routine  of 
handling  trigonometric  terms.  When  they  are  both  used  each  serves  as  a 
check  on  the  other.  There  is  little  advantage  with  either  over  the  other,  so 
far  as  their  convenience  is  concerned;  the  integral  has  a  slight  advantage  in 
that  when  using  it  the  computations  are  made  with  cosines,  and  the  disad- 
vantage of  involving  u  to  the  fourth  degree  instead  of  only  to  the  third. 


78  PERIODIC    ORBITS. 

II.    DIGRESSION  ON  HILL'S  EQUATION. 

50.  The  x-Equation.  —  The  value  of  z  was  given  explicitly  in  (50),  and 
since  it  is  periodic  the  second  equation  of  (4),  after  transforming  to  the 
independent  variable  T,  has  the  form 

2+  •  •  '  ]z  =  0,  (57) 


where  a  is  a  constant  independent  of  /z,  and  61  ,  B2,  .  .  .  are  periodic  functions 
of  T,  having  in  this  case  the  period  TT.  Since  the  period  can  always  be  made 
2ir  by  a  linear  transformation  on  the  independent  variable,  we  shall  suppose 
for  the  sake  of  uniformity  that  the  period  is  2ir. 

This  problem  belongs  to  the  class  which  was  treated  in  Chapter  I,  and 
can  be  transformed  to  the  form  considered  there.  But  it  is  now  in  the  form 
used  first  by  Hill,  and  later  by  Bruns,  Stieltjes,  Harzer,  Callandreau,  etc., 
and  because  of  its  historical  interest  and  for  the  sake  of  comparison  with 
this  earlier  work,  it  will  be  treated  directly  in  the  form  (57)  . 

51.  The  Characteristic  Equation.  —  Suppose  that  with  the  initial  con- 
ditions z(0)  =  1,  i(0)  =  0  the  solution  of  (57)  is 


and  that  with  the  initial  conditions  z(0)  =0,  i(0)  =  1,  the  solution  of  (57)  is 

X  =  *(T),  i  =  iKr), 

The  determinant 


being  equal  to  unity  at  T  =  0,  these  solutions  form  a  fundamental  set.  In 
fact,  A  is  independent  of  T  for  an  equation  of  the  form  (57),  as  was  shown  in 
§  18.  It  follows  from  the  initial  conditions  that  its  value  is  unity.  Hence 
the  general  solution  of  (57)  is 

^(T).  (58) 


where  ct  and  c2  are  arbitrary  constants. 
Now  let  us  make  the  transformation 

x  =  e"t,  (59) 

where  a  is  an  undetermined  constant.     Then  equation  (57)  becomes 

£+2a£+a2£+[a2+0lM  +  02M2+    -   -   -  ]£  =  0.  (60) 

The  general  solution  of  this  equation  is,  by  (58)  and  (59), 

M].     (61) 


HILL'S   DIFFKHKMIAL   EQUATION.  79 

We  now  raise  the  question  whether  it  is  |>ossible  to  determine  a  in  such 
a  way  that  £  shall  l>e  periodic  with  the  period  2ir.  It  follows  from  the  form 
of  (60)  that  sufficient  conditions  for  the  periodicity  of  $  with  the  period  2*  arc 


On  substituting  from  (61),  we  get,  after  making  use  of  the  initial  values  of  <p, 
j>  +  and  /, 


In  order  that  these  equations  may  have  a  solution  other  than  the  trivial  one 
r,  =c,  =  0,  the  determinant  of  the  coefficients  of  c,  and  c,  must  vanish;  or, 


„*« 


(63) 

Since  A  is  equal  to  unity,  equation  (63)  is  a  reciprocal  equation,  and  becomes 

D=(e*a')t-[<f>(2ir)+v,(2ir)]et*w+l=Ql  (64) 

of  which  t  he  roots  are  e2a'*  and  e-la<*.     (This  is  not  the  o,  of  §44.) 

When  the  value  of  e3a'r  which  satisfies  (64)  is  substituted  in  (62)  the 
ratio  of  c,  to  c,  is  determined.  Then  equations  (61)  give  £'"  and  £(l> 
periodic  with  the  period  2r.  We  get  a  second  solution  £<2)  and  £<2)  by  using 
the  other  root  e"2*1'.  The  £(l)  and  £(2)  will  each  carry  an  arbitrary  factor. 
We  shall  determine  these  factors  so  that  £("  (0)  =  £{2)  (0)  =  1,  and  multiply 
the  solutions  by  arbitrary  constants  at  the  end. 

Consider  equation  (64).  If  \<p(2ir)+  ^(2ir)|<2,  o,  is  a  pure  imaginary; 
if  <p(2*)  +^(2ir)  >  2 ,  a,  is  real ;  and  if  <p(2ir)  +\f/(2ir)  <—2,  a,  is  complex.  In 
the  first  case  x  remains  finite  for  all  real  values  of  T;  in  the  second  case  x 
becomes  infinite  as  r  becomes  infinite  through  real  values;  and  in  the  third, 
z  =  oo  for  T  =  OO  except  for  special  initial  conditions.  It  is  found  from  (57) 
that  <p(2ir)  =  ^(2ir)  =  cos  a  T  for  /i  =  0.  Therefore  the  part  of  a,  which  is  inde- 
pendent of  n  is  the  pure  imaginary  a  V— 1.  Suppose  a  is  not  an  integer; 
then  o,  is  a  pure  imaginary  for  all  real  values  of  n  whose  modulus  is  sufficient  ly 
small.  If  a  is  an  integer,  the  value  of  a,  for  real  values  of  M  whose  modulus 
is  small  may  be  purely  imaginary,  real,  or  complex  according  to  the  values 
of  <p(2ir)  and  ^(2*-). 

Some  of  the  more  important  properties  of  £(l)  and  £<2)  will  be  derived. 
There  are  two  particular  solutions  of  (57)  of  the  form  x  =  e°'r  £  such  that  a, 
is  a  constant  reducing  to  ±  a  V—i  for  n  =  0,  and  such  that  £  is  periodic  with  the 
l>eriod  2ir ,  viz.  ctt'r  £(1)  and  e~a'r£(2).  The  coefficients  of  (57)  by  hypothesis  are 
all  real,  and  the  9,  are  sums  of  cosines  of  multiples  of  r.  Therefore,  if  the 
sign  of  v^T  be  changed  in  a  solution  the  result  will  be  a  solution.  Suppose 


80  PERIODIC    ORBITS. 

a  and  n  have  such  values  that  at  is  a  pure  imaginary.  Then  it  follows  that 
£(1>(V—  i)=£<2>(  —  V—  i).  Similarly,  since  (57)  is  unchanged  by  changing 
the  sign  of  r,  it  follows  that  £cl>  (r)  =  £<2)  (  -  r)  .  And  finally,  since  (57)  is 
unchanged  by  changing  the  sign  of  both  v^T  and  r,  it  follows  that 
e\V=l,T-)  =  e\-^S=i,-T),  r(v^7,T)=£(2>(-V^T,-r).  Therefore 
in  the  expressions  for  £(1)  and  £<2)  the  coefficients  of  the  cosine  terms  are 
real  and  of  the  sine  terms  pure  imaginaries,  and  £(1)  and  £(2)  differ  only  in 
the  sign  of  V—i.  Hence,  writing  them  as  Fourier  series,  we  have 

£U)=  S  [CIJCOSJT+  V—  1  bjsmjr], 
£(2)  =  S  [a,;  COSJT  —  V  —  1  bj  sinjr], 

where  the  aj  and  the  bj  are  real  constants.  It  follows  from  this  that  it  is 
sufficient  to  compute  £(1). 

Any  solution  of  (57)  can  be  expressed  linearly  in  terms  ea'r  £(1>  and 
e-a,T£<2>  jj.  fonows  from  the  initial  values  of  <p,  lp,  £u>,  and  £(2)  and  the 
equations  above  that 


Since  ^a>  and  |(2)  are  periodic  with  the  period  2  TT,  and  since  their  initial  values 
are  unity,  we  have 

I  (ea«.'+e-2ft")  =  cosh  2alv  = 


But  by  (64),  e2a'T+e-2a"r  =  <p(27r)+^(27r).     Therefore  <p(2ii) 

When  a!  =  /3V  —  1  is  a  pure  imaginary,  as  it  is  in  many  physical  problems, 
we  have 

).  (65) 


This  equation  has  the  same  form  as  that  developed  by  Hill  in  his  memoir  on 
the  motion  of  the  Lunar  Perigee,  Ada  Mathemetica,  vol.  VIII,  pp.  1-36,  and 
Collected  Works,  vol.  I,  pp.  243-270.  It  is  also  derived  differently  in 
Tisserand's  Mecanique  Celeste,  vol.  Ill,  chap.  1. 

Equation  (64)  or  (65)  furnishes  a  means  of  computing  the  transcendental 
a,  because,  under  the  hypotheses  on  (57),  <p  can  always  be  found,  for  example 
as  a  power  series  in  ju,  with  any  desired  degree  of  accuracy.  Though  this 
constitutes  a  complete  solution  of  the  problem  and  there  are  no  difficulties 
in  carrying  it  out  except  those  of  the  lengthy  computations,  we  shall  find 
it  convenient  to  make  use  of  more  of  the  properties  of  equation  (57),  and 
to  find  both  cti  and  £t  otherwise. 

When  ;u  =  0  the  general  solution  of  (57)  is  known  to  be 


-ttV~tr,          rc0  =  a\/^T  [o,  e«  v/r'r  —  a2e-a%/zlr],        (66) 
where  a^  and  a2  are  arbitrary  constants. 


llll.l.'s    DIKKKHKM  I  \l.    K^I'ATION. 


81 


It  follows  from  the  form  of  :.~>7i  that*  can  l»e  expanded  as  ;i  power  serie- 
in  n  which  will  converge  for  0^r^2r  if  |M|  \8  sufficiently  small.     Hence 


(67) 


Therefore  equation  (6ii) 


cos2air-e'a'+ 


X-l 


' 

" 


2  *»<2 


X-l 


-asin2air  +  2  ^(2ir  V,  cos2ar  -«*•*  +  2  ^x( 

X-l  x-l 


=  0.      (68) 


This  e(|ii:it  ion  expresses  the  condition  that  ~»7  >liall  ha\e  a  |>eriodic  solution 
of  the  form  (59),  where  $  is  i>eriodic  with  the  period  2r.  If  it  is  satisfied 
by  a^a,,  it  is  also  satisfied  by  a  =  ot)+»'  V—  1,  where  v  is  any  integer.  Tin  •-«• 
different  values  of  o  do  not,  however,  lead  to  distinct  values  of  x.  \\  C 
-hall  use  only  those  values  which  reduce  to  ±a  V—  1  for  /u  =  0. 


52.  The  Form  of  the  Solution.  —  For 


the  principal  solutions  of  (68) 


are  a(,0)=  +(i  V—  -T  and  oa°>=  —a  V—  l.     There  are  three  cases  depending 
upon  the  value  of  «: 

Case     I.  «^0  and  2«  not  an  integer. 
Case   //.  a  7^0  and  2o  an  integer. 
Case  ///.  a  =  0  and  therefore  oj01  =  a?'  =  0. 

Case  I.  This  may  be  regarded  as  being  the  general  case,  and  is  that 
actually  found  and  discussed  by  Hill  and  later  writers  on  the  same  subject. 
It  follows  from  the  form  of  (68)  that 

#  =  P(a,M),  (69) 

where  P  is  a  power  series  in  a  and  M  which  vanishes  for  n  =  0,  a=  ±a  V—  i. 
It  is  also  easily  found  for  /*  =  0  and  a  =  *a  V  —  1  that 


=  ±  4ir  V= 


da 


which  is  distinct  from  zero  under  the  conditions  of  Case  I.    Therefore  it  follows 
from  the  theory  of  implicit  functions  that  (68)  can  be  solved  for  a  in  the  form 


where  the  series  converge  for  >  sufficiently  small.     Since  the  equation  for  a 
is  a  reciprocal  equation  in  e7",  it  follows  that  a\"  =  —  o^O'-l,  2,  ...,»). 


82  PERIODIC    ORBITS. 

If  we  substitute  either  of  the  series  (70)  in  (62),  we  shall  have  the  ratio 
of  Cj  and  c2  expressed  as  a  power  series  in  M-  If  this  result  and  equations 
(67)  are  substituted  in  (61),  £  will  be  expressed  as  a  power  series  in  n, 
converging  for  \p\  sufficiently  small,  and  carrying  one  arbitrary  constant 
factor.  We  shall  omit  the  superfix  and  adopt  the  notation 


2+  •  •  •  •  (71) 

Since  the  periodicity  conditions  have  been  satisfied,  £  is  periodic  and  it 
follows  from  its  expansibility  that  each  £,  separately  is  periodic.  Hence 
it  follows  from  this  property  and  the  initial  condition  £  (0)  =  1  that 

€<(2ir+T)-€i(T)=0  (t-0,  1,  .  .  .  oo),  j 

£0(0)  =  1,     £,(0)  =  0  (i  =  l,2,  .  .  .  oc).  J 

It  will  be  shown  when  the  solutions  are  constructed  that  these  properties 
uniquely  define  their  coefficients. 

Case  II.     In  this  case  we  find  from  (68)  f  or  n  =  0,  a  =  ±  a  V^Tf  ,  that 


Hence  if  we  let  a  —  a  V—  1+/8,  the  expression  for  D  has  the  form 

47r2/32+Cll/3M+c02M2+  •  •  •   -0,  (73) 

where  in  general  cu  and  c02  are  distinct  from  zero.  In  order  not  to  multiply 
cases  indefinitely  we  shall  suppose  that  cu  and  c02  are  distinct  from  zero  and 
that  the  discriminant  of  the  quadratic  terms  of  (73),  viz.  8  =  c2n  —  167r2c02  , 
is  distinct  from  zero.  Suppose  the  quadratic  terms  factor  into 


where  now  b^b2  since  d^O.*     Then  by  the  theory  of  implicit  functions 
equation  (73)  is  solvable  for  /8  as  converging  power  series  of  the  form 


+  •  •  •  •  (74) 

Hence  we  get  two  solutions  for  /3,  and  consequently  for  a,  as  power  series  in 
M  starting  from  the  root  a=  +a  V—  1  for  fj,  —  0.  There  are  two  similar 
ones  obtained  by  starting  from  a  =  —  a  V—i  for  n  =  0,  but  they  do  not  lead 
to  distinct  solutions  since  they  differ  from  the  former  values  by  purely 
imaginary  integers.  Then,  by  means  of  (62)  and  (61),  we  obtain  the  final 
solutions  as  before. 

There  are  other  sub-cases,  for  example  &j  =  &2,  all  of  which  can  be 
treated  by  the  theory  of  implicit  functions,  but  they  will  be  omitted. 

*Since  a  =  o,  and  a=—  c^  are  the  roots  of  (63),  it  follows  that  in  this  case  6i  =  —  b,,  and  that  they  are 
therefore  distinct  unless  bt  =  b,  =  Q. 


1     Ml  KKKKMIAI.    Kljl   Ali<>\  M 

/.»••«•  ///.      I'nder  the  conditions  of  this  case  \ve  have  for /i  =  0,  instead 

of  equation.-    till    and    »i7  .  the  solution 


Jo  =  a,,        V.=  l,        ^o  =  ^-  (75) 

Hence  equation 


D  =  e~i 


.-eta'+  2  <?x(2ir)/,  2r+  2 

X-l  X-l 

X-l  X-l 


=  0.       (76) 


As  before.  I)  can  l><  expanded  into  a  converging  |>ower  series  in  a  and 
n,  and  for  n  =  Q  the  principal  solutions  of  D  =  0  are  o,  —  o,  =  0.  We  find 
from  (76)  for  M  =  a  =  0  that 


In  general  ^,(2ir)  is  distinct  from  zero,  and  when  it  is  we  know  from  the 
theory  of  implieit  functions  that  (76)  can  be  solved  for  a  in  the  form 

•  •  •       (77) 


Since  a,  and  as  differ  only  in  sign  the  a',1"  are  all  zero.  After  a,  and  a,  have 
been  determined,  we  obtain  the  final  solutions  as  before,  except  now  the 
series  proceed  in  powers  of  N/P  instead  of  in  powers  of  //. 


53.  Direct  Construction  of  the  Solutions  in  Case  I.  —  On  substituting 
the  first  of  (70)  and  (71)  in  (60)  and  equating  the  coefficients  of  the  several 
powers  of  n  to  zero,  we  obtain 


,7sj 


the  left  members  of  all  the  equations  being  the  same  except  for  the  subscripts, 
and  the  first  terms  on  the  right  being  the  same  except  for  the  superscripts 
on  a,  .  There  is  a  similar  set  of  equations  defining  the  other  solution,  which 
differ  from  these  only  in  the  sign  of  V—\. 

Consider  the  solutions  of  (78)  subject  to  the  conditions  (72).     The 
general  solution  of  the  first  equation  is 


where  b™  and  6^  are  arbitrary  constants  of  integration.     Since  in  this  case 
2  a  is  not  an  integer,  it  follows  from  (72)  that 

£o=l.  (79) 


84  PERIODIC    ORBITS. 

The  right  member  of  the  second  equation  of  (78)  now  becomes  a  known 
function  of  r.     When  the  left  member  of  this  equation  is  set  equal  to  zero,  its 

general  solution  is 

e-2aV-lT.  (80) 


Now  regarding  &<"  and  &£"  as  variables,  according  to  the  method  of  the 
variation  of  parameters,  and  imposing  the  conditions  b™  +  b™  e~2a"/~~1T  =0 
and  that  the  second  equation  of  (78)  shall  be  satisfied,  we  obtain 

fQ-i  \ 


Let  the  constant  part  of  Bl  be  d^.  Then  in  order  that  &"*  shall  not 
contain  a  term  proportional  to  r,  which  would  make  (80)  non-periodic,  we 
must  impose  the  condition  

(1)        V  —  1  Q!  (Q<)\ 

ft,     —  •  ^o^^ 

The  integrals  of  sines  and  cosines  are  cosines  and  sines  respectively,  and 
Therefore  it  follows,  when  (82)  is  satisfied,  that 


-r  P--r  -? 

-'  +     — 2  sin 


,  (83) 

where  P:  and  Qj  are  periodic  functions  of  r  having  the  period  2ir,  and 
BI"  and  B"5  are  the  constants  of  integration.     Since  2  a  is  not  an  integer, 
/  —  4a2  can  not  vanish  and  there  are  no  terms  with  infinite  coefficients. 
On  substituting  (83)  in  (80)  and   imposing   the   conditions  (72),  we   get 

*i  =  P»  (r)  +  Q1  (r}  ~  Pl  (0)  -  Q,  (0)  .  (84) 

It  is  easy  to  show  that  ah1  succeeding  steps  of  the  integration  are 
entirely  similar.     The  differential  equation  for  the  coefficient  of  M*  is 


where  Ft(r)  is  an  entirely  known  periodic  function  of  T  after  the  preceding 
steps  have  been  taken.  The  general  solution  of  the  left  member  of  this 
equation  set  equal  to  zero  is  the  same  as  (80)  ,  except  that  £  has  the  subscript 
i,  and  6,  and  b2  have  the  superscript  i.  The  equations  analogous  to  (81)  are 

/>">—  „<*>  —  ^       *~ 

61  -    "L"1'     "2^ 
If  we  represent  the  constant  part  of  Ft  by  dt  ,  we  must  impose  the  condition 


in  order  that  the  solution  shall  be  periodic.     Then  integrating,  substituting 
in  the  equation  analogous  to  (80),  and  imposing  the  conditions  (72),  we  get 


where  P4(r)  and  Q,(r)  are  periodic  with  the  period  2?r.     Thus  the  general 
step  in  the  integration  is  in  all  essentials  similar  to  the  second  step. 


HILL'S  DIH  i  \i.  EQUATION.  85 

54.  Direct  Construction  of  the  Solutions  in  Case  II.  —  Since  in  this  case 
the  solutions  arc  al.-u  in  ^rnrral  oVvrlopahlr  as  power  series  in  ft,  we  start 
from  equations  (78).  The  general  solution  of  the  first  equation  is 


Since  2o  is  an  integer,  £„  is  periodic  for  all  values  of  6[w  and  b™. 

In  this  case  it  is  convenient  in  the  computation  to  impose  the  initial 
condition  £  (0)  =  1  instead  of  {(0)  =  1,  whence 

£0(0)  =  1,  £,(0)=0         (i-1,  ...«). 

1  It  ace  we  have  for  the  solution  at  the  first  step  of  the  integration 


where  b™  is  a  constant  which  will  be  determined  at  the  next  step. 
The  second  equation  of  (78)  now  becomes 


(85) 


(86) 
The  equations  analogous  to  (81)  are  in  this  case 


(87) 


In  order  that  £,  shall  be  periodic  the  right  members  of  these  equations  must 
contain  no  constant  terms.  Hence  we  must  impose  the  conditions 

6r  =  0,         (88) 

where  d,  is  the  constant  part  of  0,  ,  and  where  5,,  and  &M  are  the  constant 
parts  of  \^~l  eie~uv=lT/2a  and  v^T  01eta>/:r7r/2a  respectively.  If  6t  is 
an  even  function  of  T,  then  $„,  =  $»,  and  if  it  is  an  odd  function,  5,,=  —  5W  . 
Equations  (88)  express  the  conditions  that  the  right  member  of  (86)  shall 
contain  no  terms  independent  of  T,  or  which  involve  T  only  in  the  form 
e-i«%/=7T  This  is  only  an  expression  for  the  fact  that  in  order  that  the 
solutions  shall  be  periodic  the  right  member  of  the  differential  equation 
must  not  contain  terms  of  the  type  obtained  by  integrating  the  left  member 
set  equal  to  zero. 

Upon  eliminating  6J0'  from  (88),  we  get 

[20  V^T  C+d,]  [2av^T  a™  -d.1-401  «„*»-(), 
of  which  the  solutions  are 

^^  (89) 


The  two  values  of  a"'  are  distinct  unless  «„  5M-|-d?/4a*  =  0.     They  will  not 
be  equal  to  zero  except  for  special  values  of  the  coefficients  of  the  differential 


86  PERIODIC    ORBITS. 

equations,  and  we  shall  assume  here  that  they  are  distinct.  This  was  the 
case  treated  in  §52,  Case  II,  and  when  the  two  values  of  a"'  are  equal  the 
solutions  may  be  developable  in  a  different  form.  After  a"'  has  been 
found,  &i0)  can  be  obtained  at  once  from  either  of  equations  (88) .  There  is 
a  difficulty  only  if  52l  =  522  =  0,  when  one  solution  for  b(°}  becomes  infinite; 
but  in  this  case  we  impose  a  different  initial  condition  on  £. 

After  satisfying  equation  (88),  the  integrals  of  (87)  have  the  form 


2    > 


where  Pl  (T)  and  Qt  (T)  are  periodic  functions  of  r,  and  Bf  and  B^  are  unde- 
termined constants  of  integration.  These  results  substituted  in  equation 
(80)  give,  after  applying  the  condition  ^(0)  =  0, 


2^-,  (90) 

where  B™  is  so  far  undetermined. 

It  is  necessary  to  carry  the  integration  one  step  further  in  order  to 
prove  that  the  general  term  satisfying  the  periodicity  condition  and  the 
initial  condition  can  be  found.  The  differential  equation  for  the  coefficient 
of  /i2  is 


-2aV~l  al"  B^-6,  B(»+F2(r),  I 

where  af  and  B("  are  undetermined  constants,  and  where  F2(r)  is  an  entirely 
known  periodic  function  of  T. 

In  the  case  under  consideration  the  equations  corresponding  to  (88)  are 


-a®  +  2a  V^ 


where  d.2  and  52  are  known  constants  depending  on  F2  .  The  unknowns 
aJ2>  and  B™  enter  (92)  linearly,  and,  by  means  of  (88),  the  determinant  of 
their  coefficients  becomes  —  4aV^-7  a"',  which,  by  hypothesis,  is  distinct 
from  zero.  Therefore  a"'  and  B\"  are  uniquely  determined  by  these  equa- 
tions. When  equations  (92)  are  satisfied,  the  solution  of  (91)  satisfying  the 
initial  condition  is 


which  has  the  same  form  as  (90).  Therefore  the  next  step  can  be  taken  in 
the  same  manner.  Thus  it  is  seen  that  the  process  is  unique  after  the  choice 
of  the  sign  of  aj])  is  made,  and  in  this  way  two  solutions  which  satisfy  the 
periodicity  and  initial  conditions  are  obtained. 


III1.  1.'-    miKKHKM  I  AI.    K.I  I    \!ION.  87 

55.  Direct  Construction  of  the  Solutions  in  Case  III.  —  In  this  case  t  he- 
solutions  were  proved  to  have,  in  ^eiientl.  the  form 

«  =  *„+*,  v^+&M+          •   ,  a  =  0+a"VM+a«M  +   •   •   •   •     (94) 

Substituting  these  equal  ions  in  (60),  we  have  for  the  term  independent  of  Vji 

u=o, 

of  which  the  solution  satisfying  the  conditions  (72)  is 

«.=  !.  (95) 

The  differential  equation  which  the  coefficient  of  \//I  must  satisfy  is 


and  the  solution  of  this  equation  which  satisfies  the  conditions  (72)  is 

£,  =  0,         o(l)=  an  undetermined  constant.  (96) 

The  differential  equation  which  defines  the  coefficient  of  M  is  then 
-  -2a">     -.'»'«-«-  -amf-«  - 


When  the  left  member  of  this  equation  is  set  equal  to  zero,  its  general  solution 
i.  found  to  be  ,  (97) 


where  6(,2)  and  6™  are  the  constants  of  integration.  On  making  use  of  the 
variation  of  parameters  and  imposing  the  condition  that  the  differential 
equation  shall  be  satisfied,  we  find 

&}"=  -[(«'")'+*,],  4?-  +[(an>m]  r.  (98) 

In  order  that,  when  the  first  of  (98)  is  integrated  and  the  result  substituted 
in  (97),  there  shall  be  no  term  proportional  to  T*,  the  condition 

(am)'+d,  =  0  (99) 

must  be  imposed,  where  dt  is  the  constant  part  of  0,  .  This  equation  deter- 
mines two  values  of  a(l)  which  differ  only  in  sign,  and  they  are  reals  or  pure 
imaginaries  when  the  coefficients  of  0,  are  real. 

Since  there  are  no  more  arbitraries  available  in  (98),  no  more  conditions 
can  be  satisfied.     The  first  equation  of  (98)  gives  rise  to  integrals  of  the  type 


The  second  of  (98)  gives  rise  to  the  corresponding  and  associated  integrals 


Hence,  imposing  the  condition  (99),  integrating  (98),  and  substituting  the 

results  in  (97),  we  have 

*,  =  P,(r)-P,(0),  (100) 

where  P,(T)  is  periodic  and  a(1)  is  as  yet  undetermined. 


88  PERIODIC    ORBITS. 

In  determining  the  coefficient  of  MV!,  the  equations  for  &J3)  and  &£"  corre- 
sponding to  (98)  are  found  to  be 


where  <f3(r)  is  a  periodic  function  of  T.     If  d3  is  the  constant  part  of  tf>3  ,  we 
must  impose  the  condition 


which  uniquely  determines  a<2)  if  aa)  is  distinct  from  zero,  as  it  is  in  general. 
If  a(1)  =  0  the  expansion  may  be  of  another  form,  for  this  is  an  exceptional  case 
in  the  existence  discussion,  and  it  is  necessary  to  go  to  higher  terms  of  the 
differential  equation  to  determine  the  character  of  the  solution.  But 
limiting  ourselves  here  to  the  case  where  au)  is  distinct  from  zero,  the  solution 
is  carried  out  as  in  the  preceding  step.  All  the  succeeding  steps  are  the 
same  except  for  the  indices  and  the  numerical  values  of  the  coefficients. 

III.    SOLUTION  OF  THE  X  AND  Y-EQUATIONS  FOR  THE 
SPHERICAL  PENDULUM. 

56.  Application  to  the  Spherical  Pendulum.  —  On  transforming  from  t  to 
T  as  the  independent  variable  in  the  second  equation  of  (4),  and  making  use 
of  (50),  we  get 


«,-«,)  cos  2r], 


(101) 


Ctl       0.3  fli       0.3 

Obviously  a  will  not  in  general  be  an  integer.  It  will  be  shown  that  the 
only  integral  value  it  can  have  in  the  problem  of  the  spherical  pendulum  is 
unity.  Suppose  a  equals  the  integer  n.  In  this  case  the  second  of  (101)  gives 


It  was  shown  in  §43  that  in  the  problem  of  the  spherical  pendulum  at  is 
positive  and  a3  is  negative.  Therefore  n-  must  be  unity  or  zero.  In  the 
former  case  we  have  a,=  —  a3,  which,  because  of  the  inequalities  satisfied  by 
a,  and  a3 ,  can  be  true  only  if  a1=  +1,  a3=  —  I.  This  is  the  special  case  of 
the  simple  pendulum.  If  n  =  0  we  have  4a:=  —  2a3,  which,  because  of  the 
inequalities  to  which  a,  and  a3  are  subject,  can  not  be  satisfied.  Therefore  a 
is  not  an  integer  and  the  equations  can  be  integrated  by  the  methods  of  §53. 
Upon  emitting  the  subscript  on  the  a  in  ear,  so  as  not  to  confuse  it  with 
a, ,  cu  ,  a,  defined  in  §43,  it  is  found  by  actual  computation  that 


4V(aI-a3)(2a1+a3)  ' 

.   fl  _  1  t    X  ft         \      ft  _   I  . , 

V--1  Sin2r, 


1111.    M'HKIUCAK    1'KMH  I.I   M. 


The  other  solution  is  found  from  this  one  simply  l>y  eh.-inninn  the  sign  of 
\  —1.      Hence  I  lie  general  solution  is 


fV2v  2a, 

,          4V 

-1) 


"1 
'  J  V- 


(102) 


1  1  can  he  shown  from  the  properties  of  0,  ,  0,  ,  .  .  .  and  the  method  of 
constructing  the  solutions,  that  the  coefficients  of  n,  in  £(l)  and  £(2)  are  cosim  •> 
and  sines  of  even  multiples  of  T,  the  highest  multiple  being  2j. 

It  follows  from  the  form  of  equations  (102)  that,  for  real  initial  condi- 
tions, A  and  B  must  be  conjugate  complex  quantities,  2A  =  At  —  V—  I  At, 
and  2  B  =  At  +  V^~iAs  .  Hence  the  solution  takes  the  form 


(103) 


x,  =  l-          -^r(cos2T-l)/i  +  (cosines)  / 
$(a.t-\-at) 

V2  V(a  -«J  (2 


where  /I,  and  ^4,  are  arbitrary  constants. 

Since  the  second  and  third  equations  of  (4)  have  the  same  form,  the 
solution  of  the  latter  can  differ  from  that  of  the  former  only  in  the  constants 
of  integration.  Therefore 

y  =  Bl(xl  cosXr  —  xt  sin \r]-\-Bj [x,  sinXr+x,  cosXr]. 

The  constants  Al ,  Aa,  B, ,  and  Bt  are  subject  to  the  conditions  that 
equations  (2),  the  first  equation  of  (4),  and  the  relation 

xx+yy+zz=Q 

shall  be  satisfied.  This  leaves  one  arbitrary  which  may  be  used  to  dispose 
of  the  orientation  of  the  xy-&xcs  at  r  =  0.  Let  the  axes  be  chosen  so  that 
x  =  0  at  T  =  0.  Then ,  since  z  also  vanishes  at  T  =  0,  we  have  from  this  equation 
and  the  values  of  x  and  y  above 

— x,sinXT],  j/  =  /?,[x,  sinXr+x,  cosXr].        (104) 


90  PERIODIC    ORBITS. 

Making  use  of  (104),  it  is  found  from  (2)  and  the  first  of  (4),  that  at  r  =  0, 


-(a,-a3)[X+X2(0)T2 

Therefore,  the  solution  is  completely  determined  when  the  positive  directions 
on  the  x  and  ?/-axes  are  chosen.  The  well-known  properties  of  the  motion 
can  easily  be  derived  from  equations  (104). 

The  variables  x  and  y  always  oscillate  around  their  initial  values  since 
they  are  made  up  of  terms  which  are  the  product  of  two  periodic  functions 
that  are  always  finite.  Since  the  period  of  xl  and  x2  is  TT,  the  solutions  are 
periodic  and  the  curves  described  by  the  spherical  pendulum  are  re-entrant 
provided  X  is  a  rational  number.  Let  the  period  of  xl  and  x2  be  PI  =  TT, 
and  that  of  sinXr  and  cosXr  be  P2=27r/X.  Then,  when  Pl  and  P2  are 
commensurable, 

P?  =  2  =  2j 

Pl      X    "  p  ' 

p  and  q  being  relatively  prime  integers.  Hence  the  least  common  multiple 
of  the  two  periods  is 

^  =2?7r.  (105) 


In  the  period  P  the  variables  z,  xl  ,  and  x2  make  2q  complete  oscillations, 
and  sin  Xr  and  cos  XT  make  p  complete  oscillations.  In  the  independent 
variable  T  the  period  of  z,  xt  ,  and  x2  is  independent  of  M,  but  P2  is 
a  continuous  function  of  /JL.  In  the  original  independent  variable  t  both 
periods  are  continuous  functions  of  t.  But  in  either  variable  the  period  P 
is  a  discontinuous  function  of  n,  being  finite  only  when  the  ratio  of  Pl  to 
P2  is  rational.  It  is  seen  from  the  solution  expressed  in  terms  of  T,  in 
which  P,  is  constant  with  respect  to  n,  that  this  ratio  fills  a  portion  of 
the  linear  continuum,  and  therefore  that  onhy  exceptionally  is  it  rational. 

57.  Application  to  the  Simple  Pendulum.  —  Since  the  problem  of  the 
simple  pendulum  is  a  special  case  under  that  of  the  spherical  pendulum,  it 
can  be  treated  by  the  same  methods.  Of  course,  it  is  not  advisable  to  do 
so  in  practice,  for  x  and  z  must  satisfy  the  relation 

z2  +  z2=f,  (106) 

from  which  x  can  be  found  when  z  has  been  determined. 

Before  discussing  the  properties  of  x  we  must  find  the  expression  for  z 
in  this  case.  Since  in  the  simple  pendulum  z  always  passes  through  the 


I  Hi:    -IMIKKH  Al.    I'KNDl  I. I'M.  91 

value  — /.  it  follows  that  a3=  —  I.  I'Voiii  the  fact  that  z'  -  II  \\hcii  z=—l, 
we  get,  by  (5),  c,  =  f,  and  z'  =  0  for  z=+l.  Hence,  in  the  case  of  the 
simple  pendulum  we  have  from  (50)  and  (101) 


z  =  -1+1(1  -cos  2T)M+{(l-cos4T)M*+ 


a, 


0,=J(l+2cos2T), 


]x  =  0. 


(107) 


In  the  expression  for  z  the  coefficient  of  each  power  of  n  separately 
vanishes  at  T  =  0  and  is  a  sum  of  cosines  of  even  multiples  of  r.     Therefore 


contains  sinV  as  a  factor.  The  parameter  n  is  also  a  factor.  From  the 
relation  M(!  —  cos2r)  =  2/x  sinV,  it  follows  that 

x=±VF^*  (108) 

is  expansible  as  a  power  series  in  VM,  containing  only  odd  powers  of  v//u. 
It  is  easy  to  show  that  the  coefficient  of  (V/f)tl+l  is  a  sum  of  sines  of  odd 
multiples  of  T,  the  highest  multiple  being  2t  +  l.  We  find  directly  from 
the  second  line  of  (107)  and  from  (108)  that 

(109) 

It  follows  from  (109)  that  the  last  equation  of  (107)  has  a  solution  of 
the  form 

(110) 


where  the  xw+1  are  periodic  with  the  period  2*  instead  of  T.  and  where 
*n+i(0)  =  0.  It  is  not  possible  to  determine  completely  the  constants  of 
integration  from  these  conditions,  for  if  (110)  is  a  solution,  then,  since  the 
last  equation  of  (107)  is  linear,  (110)  multiplied  by  any  power  series  in  p 
having  constant  coefficients  is  also  a  solution.  For  example,  we  have  for 
the  determination  of  x,  and  x, 

i.+x^O,  xl+xJ=-|(l+2cos2T)x,) 

the  solutions  of  which,  satisfying  the  conditions  x,(0)  =x,(0)  =0,  are 

n 

x,  =  CisinT,  *t  =  c»  sinr+  —  c,  sinSr, 

where  r,  and  r,  are  undetermined.  This  indeterminateness  continues  as  far 
as  the  solution  is  carried,  unless  additional  conditions  are  imposed. 


92  PERIODIC    ORBITS. 

The  value  of  x  at  T  =  0  is  an  infinite  series  in  VM  whose  general  term  is 
not  easily  obtained;  but,  from  the  fact  that  z(ir/2)  =  a.2  =  —  Z+2/ju  and  from 
equation  (108),  we  get 


(111) 
On  determining  ct  and  c3  by  these  conditions,  we  find 

x  =  ±  if  [2  sinr]  MV'+  |[-5sinT+3sin3T]Mv'+  •  •  •}' 

agreeing  with  the  direct  computation  (109). 

We  may  also  consider  the  last  equation  of  (107)  from  the  standpoint  of 
the  general  theory  of  linear  differential  equations  having  periodic  coefficients. 
From  the  fact  that  the  part  of  the  coefficient  of  x  which  is  independent  of 
/x  is  unity,  it  follows  that  the  solution  of  this  problem  belongs  to  Case  II. 
Since  there  is  one  solution  which  is  periodic  with  the  period  2ir,  the  values 
of  a  are  independent  of  /j.  and  are  simply  ±  V—  l.  We  have  here  the  case  in 
which  the  two  values  of  a  not  only  differ  by  an  imaginary  integer  for  /*  =  0, 
but  for  all  values  of  /*.  It  follows  from  §21  that  in  this  case  the  second 
solution  is  either  r  times  a  periodic  function  or,  for  special  values  of  the 
coefficients  of  the  differential  equation,  a  periodic  function.  In  the  problem 
of  the  simple  pendulum  the  second  solution  is  T  times  a  periodic  function, 
and  is  most  simply  found  by  integrating  the  last  equation  of  (107)  with 
the  initial  conditions 


If  we  make  the  transformation  x  =  eaT£,  then  the  last  equation  of 
(107)  becomes 


(112) 


-^(5+16cos2T+4cos4r)M2+ 


We  shall  integrate  this  equation  and  determine  a  so  that  £  shall  be  periodic 
with  the  period  2w.  The  chief  point  of  interest  will  be  that  a  will  be  inde- 
pendent of  n  so  far  as  the  work  is  carried. 

The  equations  corresponding  to  (85)  and  (86)  are 


t   — 
SO 


-  f  (1+2  cos2T)  (26™+  V- 


(113) 


THK    Sl'IIKUICAI.    I'KNOULUM.  93 

The  conditions  that  the  solution  of  the  second  of  these  equations  shall  be 
periodic  are 

-  2v/-  i  CC-  f  &10)-  f  v  -  1  =0,         -a',"-  f  &10'-  f  V-l=(},      (114) 
of  which  the  solutions  are 


««>  =  „,  &»>  =  -v-i.  (115) 

Hence,  we  see  by  direct  computation  from  the  differential  equations  that  in 
this  problem  a  is  independent  of  /*  up  to  ju3  at  least. 

58.  Application  of  the  Integral  Relations.  We  now  return  to  the  con- 
sideration of  the  problem  of  the  spherical  pendulum.  Since  z  and  x  have 
been  determined  the  value  of  y  can  be  found  from  the  first  equation  of  (4). 
But  it  will  be  noticed  that  in  this  work  no  explicit  use  has  been  made  of  the 
integral  (2).  Now  x,  y,  and  z  must  satisfy  the  differential  equations,  given 
in  the  last  three  equations  of  (4),  and  the  integral  relations 


x> 


(116) 


Oi  —  O» 

xx+yy+zz=Q,  xy-yx=ct, 

where  the  second  equation  is  determined  from  (2)  by  changing  the  inde- 
pendent variable  from  t  to  T;  the  third  equation  expresses  the  fact  that  the 
motion  must  be  along  the  surface  of  the  sphere;  and  the  fourth  equation  is 
obtained  from  the  second  and  third  equations  of  (4),  and  expresses  the  fact 
that  the  projection  of  the  areal  velocity  on  the  zj/-plane  is  constant. 

The  solutions  of  the  second  and  third  equations  of  (4)  have  been  shown 
to  have  the  form 

x  =  a,  e^.+a,  e~ar^  ,  y  =  bv  e*Ttt+bt  e-*T!-,  ,  (117) 

where  a,  ,  a2,  6,  ,  and  6,  are  arbitrary  constants,  £,  and  £2  are  jwwer  series  in  n 
and  are  periodic  with  the  period  T,  and  o  is  a  pure  imaginary  which  is  also  a 
power  series  in  n,  but  is  not  an  integer.  Moreover,  £,  and  £,  are  conjugate 
complex  quantities. 

If  we  make  use  of  (117),  the  first  equation  of  (116)  becomes 

(aH-6^?eiar+(^+6J)^e-w'-|-2(ala,+616,)fl{2  =  r-z1.          (118) 

Now  it  has  been  shown  that  z2  is  expansible  as  a  power  series  in  n  and  that 
it  is  periodic  with  the  periodic  T. 

Before  proceeding  further  we  shall  prove  a  lemma.     Suppose  there  is 
given 


Suppose  the  <PI(£)  are  not  identically  zero,  that  they  are  periodic  with  the  period 
2ir,  and  that  no  two  of  the  a,  arc  equal  or  differ  by  an  imaginary  integer. 


94  PERIODIC    ORBITS. 

Then  let  e2ffjw  =  kj.     Suppose  for  t  =  t^  that  p,(O  ,  .  .  .  ,  <?„,  (0  are  distinct 
from  zero.     Then  we  have 

TT)  =2^,6^(0^     =0, 
=      a,e"V  (Qk,     =  0, 


It  follows  from  these  equations  that  either 

«i  =  a2  =    '  /  '    =a»,  =  l 
or 

1       ,1       ,  •  •  •  ,  1 


=  0. 


Under  the  hypotheses  on  the  a}  this  determinant  is  distinct  from  zero. 
Therefore  at=   •  •  •   =aBl  =  0. 

On  taking  another  i  for  which  some  of  the  remaining  <p}  do  not  vanish, 
we  prove  similarly  that  their  coefficients  are  zero.  Continuing  thus  we 
reach  the  conclusion  that 

a;  =  0          (.7  =  1,  .  .  .  ,  n). 
Upon  applying  these  results  to  (118),  we  get  the  relations 


Therefore 

61==fcv/^To1,  62==t\/^la2,  (119) 

from  which  we  get  either 


2  =       or 

according  as  the  same  sign  or  the  opposite  signs  are  used  in  front  of  V  —  1 
in  (119).  It  follows  from  (118)  that  in  the  former  we  have  the  trivial 
case  z=  ±/.  Hence  we  shall  take  (119)  with  opposite  signs. 

The  constants  cl  and  c2of  equations  (7),  upon  which  n  depends,  arise  in 
energy  integrals  and  are  independent  of  the  orientation  of  the  x  and  ^/-axes. 
Consequently  we  may  take  the  axis  so  that  y  (0)  =  0  without  affecting  the 
work  up  to  this  point,  and  from  this  it  follows  that 

6,=  -  62  =  V^l  a,  =  V^T  a2.  (120) 

As  a  consequence  of  (119)  and  (120),  equation  (118)  becomes 

4a;$1$,  =  Z2-*2.  (121) 


IIIK    Sl'HKIUt  -AL    1'KNDULUM.  95 

Let  us  suppose  that  z  has  been  computed  and  that  we  wish  the  coefficients 
of  £,  and  £2  .     These  quantities  have  the  form 


"'  COS2T+  V^l  6J"  sin2r]  M  + 

-f   •  •  •  +a^co82jr 
+  v-  1  &i')8in2T     -f  •  •  •  +  V=i  6JV  sin2  j 

;"  COS2T-  V=i  6"'  sin2r]  M+ 
+  •  •  •  +aJ'/cos2./Y 


cos2T     4-  •  •  •  +cj'/  cos2jV] 
where  the  aJV  and  c",'  satisfy  the  relations 

+  •  •  •  +C-0, 


(122) 


w-i,...«). 

The  constant  coefficients  of  these  solutions,  as  determined  from  the 
differential  equations,  are  expressed  in  terms  of  a,,  a,,  and  p,  and  therefore 
we  must  express  (116)  in  terms  of  the  same  parameters  in  order  that  it  may 
be  possible  to  compare  these  results  with  those  obtained  from  the  integrals. 
We  find  from  (5),  (7),  and  the  first  of  (10)  that 

o:-oJ)M.  (124) 


While  the  constant  a,  is  arbitrary  in  the  solution  of  the  differential 
equations,  it  must  be  subjected  to  the  condition  that  the  pendulum  shall 
move  on  the  sphere  whose  radius  is  /.  This  condition  may  very  well  make  it 
a  power  series  in  n;  hence  it  must  be  expressible  in  the  form 


In  fact,  we  find  from  (121)  at  r  =  0,  upon  making  use  of  (50)  and  (124),  that 

(125) 


o 

If  we  substitute  (122),  (124),  and  (125)  in  (121),  we  get  results  of  the 
form 

'+  '  '  '  '  (126) 


Since  this  equation  is  an  identity  in  n,  we  have 

F,=G,  u-o,  ...»).  (127) 

By  hypothesis  z  has  been  determined;  therefore  the  G,  are  fully  known. 
It  follows  from  (122)  that  the  F,  and  the  G,  have  the  form 


;'/  COS2./V, 


96  PERIODIC    ORBITS. 

where  the  B(0n  ,  .  .  .  ,  B^  are  known  constants.    Since  equations  (127)  are 
identities  in  r,  we  have 

A$  =  B%  (i  =  0,  .  .  .  ,j-j  =  0,  ...  oo).       (129) 

On  substituting  (122)  in  the  second  of  (116),  we  get  from  this  integral 


a2 


01  —  as 
When  we  reduce  this  equation  by  (120)  and  (121),  we  obtain 


].    (130) 


Now,  on  substituting  (122)  and  the  series  for  z  in  this  equation,  we  get  an 
expression  of  the  form 


From  the  fact  that  this  expression  is  an  identity  in  /z,  it  follows  that 

H,  =  K,  O'=0,  ...oo).  (131) 

The  K}  are  known  except  for  the  expansions  of  a.     The  constant  K0  involves 
(a(0>)2,  and  the  K,(j=\,  .  .  .  oo)  involve  the  au)  linearly. 

On  referring  to  (122),  we  see  that  the  H  t  and  the  Ks  have  the  form 


K  ,  =  &»+&»  cos2r+   •  •  •   +D$  cos2;r.         j 


Since  (131)  are  identities  in  T,  it  follows  that 

€%=&»  (i=o,  ...,;;  j=0,  ...  oo).     (133) 

It  will  now  be  shown  that  equations  (123),  (129),  and  (133)  determine 
uniquely  the  a£V  ,  6",',  a"1,  a0'  (j>0),  in  the  order  of  increasing  values 
of  j  when  z  and  8  are  known.  To  do  this  it  is  necessary  to  develop  the 
explicit  forms  of  (129)  and  (133)  by  reference  to  equations  (121)  and  (130). 
It  is  necessary  to  eliminate  a2  and  I2  from  their  right  members  by  equations 
(124)  and  the  first  of  (10).  When  j  =  0,  we  get  from  (121)  and  (130) 

4(a[0))2=-2a3(a1 


0=-(o(0))2[2o3(a1 


4  a3(2ai+a3)(ai -fas) 


(134) 


01—03 

The  first  of  these  equations  determines  aJ0>  except  as  to  sign.  The  sign  of  af 
depends  upon  which  is  taken  as  the  positive  end  of  the  z-axis.  The  second 
equation  gives 

(o(0)y2  =  -  a2  =  -  2(2ai+a3) ,  (135) 

01  —  as 

agreeing  with  the  result  in  (101). 


I  UK    -IMIKHI.   Al.    I'KMM   l.l   \! .  97 

When./=  1.  we  find  from     lL'1  i  and     l.SOi  that 

I        B(o  -r-8<X"  =  -  io;-^)-2a,r;l>  = /; 

I        ^  fO  0-2atc(t"  =  Bitt\ 

'       =  0  =  -  4  a  "'  a,  ( a,  +  a,)  a'"  -  (a"»)1  (aj  -  oj)  -  2  (a(0))s  a,  ri" 

13li 


Oi  —  n 

C?»->-16(af    «A  -\b™=-2(*m 


ttl  —  Ol 


!<•  which  we  must  add  the  lir-t  equation  of  (  123)  for>=  1.  The  unknown.- 
in  the>e  five  c«  |iiations  are  <",  a1,",  a"',  a1",  and  &i",  which  enter  linearly. 
The  second  equation  determine-  <;_.'  ;  then  f^"  is  found  from  the  first  of  (123); 
then  the  first  of  (136)  defines  «;",  while  a("  and  &"'  are  jjiven  l»y  the  last  two 
equation.-  <>l"  136). 

We  shall  apply   (136)  in  computing  the  first    terms  of  the  solutions. 
We  find  from  (49)  and  (50)  that 

*,-£,  ci"=-C=  !(«,-«,).  (137) 

Upon  substituting  in  (136)  and  solving  these  equations  and  the  first  of 
(123)  for  oj",  <C,  <,  a'",  and  6i"  in  order,  we  get 


-(tti-cu)\/ai-htt, 
4  x  -2a, 

V2(ai-cu)(2o1  +  o,) 


n      = 

4  v--2a, 


I   x    a,  -a,)(2a, 


(138) 

«/O  /_  _    \  /O  _       I     _   \ 

6" 


agreeing  exactly  with  the  results  obtained  in  (102). 

For  a  general  value  of  j  the  equations  corresponding  to  (  13(5)  are 


=   -16«')sa<0)N  -1  /ftJi'+^'-DiV  (•-! j). 


(139) 


The  unknowns  in  these  e(|Uations.  after  r/  a,'  .  //,',  (/,-  =  () j—  1) 

have   been    determined,    an-   „„'  .    n,    .    ,/_.,.    a     .    /-,,    («-l,   .   .   .    ,/).       The 

A.%,   Cti,  and  /).?,'((=! j)  are  known  quantitie-  (le|>ending  upon 

t  lie  coefficients  having  smaller  numbers  for  the  su|M-rscripts.     The  j+2  equa- 
tions of  the  first  two  and  the  last  lines  define  uniquely  the  j  +  2  quantitie.- 

aml  «iV  (i  =  0 j).      The  equation   of   the   third   line  determine- 

a0',  and  the  equations  of  the  fourth  line,  the  coefficients  ///,  (t"»l,  .  .  .  J). 


98  PERIODIC    ORBITS. 

Therefore  we  have  the  interesting  result  that  in  this  problem  the  coefficients 
of  the  general  solution  can  all  be  determined  from  the  integral  relations 
alone,  the  solution  of  the  z-equation  having  been  previously  obtained  from 
another  integral  in  §  49. 

The  last  two  equations  of  (116)  are  unused  integrals.     Let  us  consider 
the  last  equation,  which  is  the  more  complicated.    By  means  of  (117),  we  get 

4  a\  a  </=T  M2-  2  a2,  V'-^l  &  &-&  &)  =ca  . 
Upon  reducing  by  (121),  this  equation  becomes 

lc3.  (140) 


The  constant  c3  will  be  a  power  series  in  ju  having  constant  coefficients. 
Hence,  on  expanding  (140)  as  a  power  series  in  n,  we  have 

I/!  M+L2/i2+  •  •  •   =  M0+M1/i+M2ju2+  •  •  •  ,  (141) 

where  the  Mt  involve  linearly  in  the  terms  independent  of  T  the  unknown 
coefficients  of  the  expansion  of  c3 .  Since  this  equation  is  an  identity  in  /u, 
we  have 

L,=M,         (j=o,  .  ...  .00). 

It  follows  from  (122)  and  (140)  that 

0 

2J>  cos2r+   •  •  •   +E(2j  cos2jr, 
"}  cos2r+   •   •  •   -\-F2j  cos2jr, 

from  which  it  follows  that 

ETO')  ft-')  (  ' A  ,,'.      « f\  rf*\  ('I  4O^ 

The  EM  and  the  F2"  (i  =  l,  .  .  .  ,  j)  are  known  functions  of  the  coefficients 
already  computed,  while  the  F(0J)  involves  the  unknown  coefficient  of  n1  in 
the  expansion  of  ca.  Consequently  equations  (142)  determine  this  constant 
for  i  =  0,  and  also  furnish  a  check  on  the  earlier  computation  of  the 
coefficients  for  i=l,  .  .  .  ,  j. 


CHAPTER  IV. 

PERIODIC  ORBITS  ABOUT  AN  OBLATE  SPHEROID. 

BY  WILLIAM  DUNCAN  MACMILLAN. 


59.  Introduction.  The  orbit  of  a  particle  about  an  oblate  spheroid  is 
not.  in  general,  closed  geometrically.  The  motion  of  the  particle  is  not, 
therefore,  in  general,  periodic  from  a  geometric  point  of  view.  But  if  we 
consider  the  <>rbit  as  described  by  the  particle  in  a  revolving  meridian  plane 
which  passes  constantly  through  the  particle,  several  classes  of  closed  orbits 
can  be  found  in  which  the  motion  is  periodic.  The  failure  of  these  orbits 
to  close  in  space  arises  from  the  incommensurability  of  the  period  of  rotation 
of  the  line  of  nodes  with  the  period  of  motion  in  the  revolving  plane.  When 
these  periods  happen  to  be  commensurable  the  orbits  are  closed  in  space  and 
the  motion  is  therefore  periodic,  though  the  j>eriod  may  be  very  great. 
Indeed,  it  seems  that  much  of  the  difficulty  in  giving  mathematical  expres- 
sions to  the  orbits  about  an  oblate  spheroid  rests  upon  the  incommensura- 
bility of  periods.  The  difficulty  arising  from  the  node  can  be  overcome  in 
the  manner  just  described,  but  elsewhere  it  is  more  troublesome. 

Orbits  closed  in  the  revolving  plane  are  considered  most  conveniently 
in  two  general  classes:  I,  Those  which  re-enter  after  one  revolution;  II,  those 
which  re-enter  after  many  revolutions.  The  existence  of  both  classes  is 
established  in  this  chapter  and  convenient  methods  for  constructing  the 
solutions  are  given.  Orbits  which  re-enter  after  the  first  revolution  are 
naturally  the  simpler  and  will  be  considered  in  the  first  part  of  the  chapter. 
Those  lying  in  the  equatorial  plane  of  the  spheroid  become  straight  lines  in 
the  revolving  plane,  and  within  the  realm  of  convergence  of  the  series 
employed  all  orbits  in  the  equatorial  plane  are  periodic.  When  the  motion 
is  not  in  the  equatorial  plane  there  exists  one,  and  only  one,  orbit  for 
assigned  values  of  the  inclination  and  the  mean  distance.  These  orbits  reduce 
to  circles  with  the  vanishing  of  the  oblateness  of  the  spheroid. 

In  considering  orbits  which  re-enter  only  after  many  revolutions  the 
differential  equations  are  found  to  be  very  complex,  and  one  would  despair 
of  ever  finding  any  of  these  orbits  by  direct  computation.  However,  a 
proof  of  their  existence  and  a  method  for  the  constructions  of  the  solutions 
are  given  by  the  aid  of  theorems  on  the  character  of  the  solutions  of  non- 
homogeneous  linear  differential  equations  with  j>eriodic  coefficients. 


100  PERIODIC    ORBITS. 

These  periodic  orbits  of  many  revolutions  involve  five  arbitrary  con- 
stants. One,  only,  is  lacking  for  a  complete  integration  of  the  differential 
equations.  The  orbits  are  all  symmetric  with  respect  to  the  equatorial  plane. 

60.  The  Differential  Equations. — The  differential  equations  of  motion 
of  a  particle  about  an  oblate  spheroid  are* 


d*x          k^Mxr,       3 , 2  2    z2+2/2-4z2  ~\      dV 


j?,  2  2    x'-^y— 4g*      ,          -i  _  d  F 
R*    ^      10° M  fl4  =  ^~' 

k*My r,      3,22  x2+t/2-422  n     ay 


3,2  2    ir-f-y— ««•     , 

w6  M         ~7F"  'J  = 

']  = 


dt*  R3  10"  fl«  dy' 

d?z          tfMzr,       3,2  23(x2+2/2)-2?2  -i      5F 


df  /?    L-      »w"  ^  J    -  a2' 


(1) 


The  symbols  employed  are  defined  as  follows: 

The  x,  y,  z  arc  rectangular  coordinates,  the  origin  being  at  the  center 

of  the  spheroid  and  the  xy-plane  being  the  plane  of  the  equator, 
k  is  the  Gaussian  constant,  b  is  the  polar  radius  of  the  spheroid, 

M  is  the  mass  of  the  spheroid,      ^  is  the  eccentricity  of  the  spheroid, 


,  n 


-I 

Since  -  -  ,  we  obtain  one  integral  of  areas,  namely 

x  dx      y  dy 


That  is,  the  projection  of  the  area  described  by  the  radius  vector  upon  the 
equatorial  plane  is  proportional  to  the  time.  We  have  also  the  vis  viva 
integral 


There  are  no  other  integrals  which  can  be  expressed  in  a  finite  number  of 
terms,  and  for  further  integration  we  are  compelled  to  resort  to  the  use  of 
infinite  series. 

It  will  be  advantageous  to  transform  the  differential  equations  to 
cylindrical  coordinates  by  the  substitutions 

z  =  arcosv,  y  =  arsmv,  z  —  aq, 


(4) 

A  &! 
10  a2 


*Moulton's  Introduction  to  Celestial  Mechanics,  p.  113. 


ORBITS   ABOUT   AN    oHI.VIK    >1'IIKH<  HI).  I'fll 

Vl'l.T   llir-r  -lll>-tittllion>  <•<  |ll:il  ioli-    i  1  .    heroine 


(a)                       r'-r(v')'-  =-  ^^-  r>~4r<?<  «»»'  I 

-'  ''5  ^ 

(6)  rr 

.         -  -0,   ,  , 

''  --*  » 


(5) 


where  tlic  accents  denote  derivatives  with  respect  to  T. 

The  integral  of  (6)  is  rV  =  r,  by  means  of  which  r'  can  be  eliminated 
from  equation    u),  and  the  equations  then  take  the  form 


v/  TI/ "/"          \i~~rif~/' 
(c) 

The  first  two  of  the.-e  equations  are  independent  of  the  third,  so  that  r  and 
7  may  he  considered  as  being  rectangular  coordinates  in  a  revolving  plane 
which  passes  always  through  the  polar  axis  of  the  spheroid  and  through  the 
particle  itself.  The  problem  is  thus  reduced  to  the  consideration  of  the 
motion  in  this  plane,  for,  when  r  is  known,  v  is  obtained  from  the  last  equation 
by  a  simple  quadrature. 

61.  Surfaces  of  Zero  Velocity. — The  velocity  integral  in  the  revolving 
plane  is 


If  we  put  the  velocity  equal  to  zero,  the  resulting  equation  represents  a 
two-parameter  family  of  curves.  For  assigned  values  of  the  parameters  c  and 
c,,  there  is  defined  a  curve  in  the  revolving  plane.  On  one  side  of  this 
curve  the  motion  is  real  and  on  the  other  side  it  is  imaginary.  For  values  of 
r,<0,  this  curve  is  closed  and  the  motion  is  real  on  the  inside.  As  the  plane 
revolves  this  curve  generates  a  surface  of  the  general  form  of  an  anchor  ring. 

For  n*  =  Q,  this  curve  belongs  to  the  ordinary  two-body  problem  and 
the  motion  is  elliptic,  parabolic,  or  hyperlwlic  according  as  c,  is  negative, 
zero,  or  positive.  Its  equation  is 

2 

(r* 
On  putting 

we  find,  by  solving  for  p,  that 


1  T 
=c,     - 


• 
-. 

102  PERIODIC    ORBITS. 

For  negative  values  of  ^  this  equation  represents  two  closed  ovals  which  do 
not  inclose  the  origin.  If  c'  c.,  =  —  1  the  oval  shrinks  upon  the  points 
p  =  —  l/c2 ,  <p  =  0  and  TT.  The  corresponding  orbit  is  therefore  a  circle  in  the 
equatorial  plane.  As  c2  approaches  zero  the  ovals  open  out  rapidly  and 
approach  the  limiting  curves 


p  = 


2  cos2  <p ' 
For  values  of  c2>0,  there  is  but  one  positive  value  for  p,  which  is 


c2 


If  c2?*Q,  none  of  these  curves  cross  the  axis  (f>  =  w/2.     But  if  c2  =  0,  we  have 
the  circle  p=  —  2/c2  inside  of  which  the  motion  is  real  when  c2  is  negative. 
For  values  of  //^O,  but  sufficiently  small,  we  can  put 

r=(p+p)  cosy,  q=(p+p)sin<p, 

and  solve  for  p  as  a  power  series  in  M2-     We  find  in  this  manner 

i  2  —  3  cos2  <p  -2    2  i 


which  is  the  correction  to  be  applied  to  the  corresponding  surface  in  the 
two-body  problem. 

I.    ORBITS  RE-ENTRANT  AFTER  ONE  REVOLUTION. 

62.   Symmetry. — On  returning  to  the  differential  equations  (a)   and 
(fe)  of  (6) ,  we  observe  that  if  we  change 

r  into+r,  q  into  —  q,  r  into  — T, 

the  differential  equations  remain  unchanged.     Hence,  if  at  some  epoch  T  =  TO 

that  is,  if  at  the  epoch  r  =  ra ,  the  particle  crosses  the  r-axis  perpendicularly, 
it  follows  from  the  form  of  the  differential  equations  that  the  orbit  is  sym- 
metrical with  respect  to  the  r-axis  and  with  respect  to  the  epoch  r  —  TO  . 
In  other  words,  r  is  an  even  function  of  T  —  TQ  ,  and  q  is  an  odd  function  of 
T-TO.  If  now  at  some  other  epoch,  T  =  TO+  T,  the  particle  again  crosses  the 
r-axis  perpendicularly,  the  orbit  is  symmetrical  with  respect  to  this  epoch  also. 
It  is  clear,  therefore,  that  the  orbit  is  a  closed  one,  and  that  the  motion  in 
it  is  periodic,  for,  at  T  =  TO  +  T  and  at  T  =  TO  —  T  it  must  have  been  at  the 
same  point  and  moving  with  the  same  velocity  in  the  same  direction.  Hence 
sufficient  conditions  for  periodicity,  with  the  period  2  T,  are 

r'(r0)  =?(TO)  =  0  ,        r'(T0+T)  =  q(r3+T)  =0  . 

From  the  areas  integral,  v'  =  c/f-,  it  follows  that  if  r  is  periodic  v  will  have 
the  form  v  =  A  (T  —  TO)  +  periodic  terms,  where  A  is  a  constant. 


OKBITS    AHon     AS     ciHI.AII.    M'HKUOID.  103 

63.  Existence  of  Periodic  Orbits  in   the  Equatorial   Plane. — In  the 

case  where  7  =  0  equations  ,t»i  reduce  to 


(8) 


The  first  of  these  equations  is  independent  of  the  second  and  can  be 
integrated  separately.  It  represents  motion  in  a  straight  line  in  the  revolv- 
ing plane.  It  admits  the  constant  solution 


r  =  r0=l,  c'  =  c; 

which  represents  a  point  in  the  revolving  plane,  or  a  circle  in  the  equatorial 
plane. 

In  order  to  investigate  the  oscillations  about  this  point  let  us  put 

r  =l+pe, 


where  p  is  a  variable  whose  initial  value  can  be  arbitrarily  assigned,  e  is  a 
parameter  corresponding  to  the  eccentricity  in  the  two-body  problem,  and  e 
is  a  parameter  to  be  determined  so  that  p  shall  be  periodic. 

On  substituting  these  values  in  (8a)  and  expanding  as  power  series  in  e, 
the  terms  independent  of  e  cancel  out,  and  it  is  possible  to  divide  through 
by  e.  The  equation  becomes 


]p=+[l-3pe+6pV-10pV+  •  •  •]« 
+  [3  -60V-  1501  M4          +---]p'e 
+1-6+100V+460V     +  •  •  ']p 

•  •  •  ]pVj 


(9) 


We  can  simplify  this  equation  somewhat  by  dividing  through  by  the  coeffi- 
cient of  p  in  the  left  member  and  then  substituting 


'+    -   '    :,  g 

The  equation  then  becomes 

=  [l-3pe+6pV-10pV+ 


+  [-6+a1]pV  +  [10+aJ]pV+ 
where 


(10) 


a,=  - 


104  PERIODIC    ORBITS. 

Equation  (10)  can  be  integrated  as  a  power  series  in  8  and  c  with  the 
initial  values 

p=-l,  P'  =  0. 

By  Poincare's  extension  of  Cauchy's  theorem,  §  §  14-16,  this  solution  con- 
verges for  values  of  5  and  e  sufficiently  small,  and  for  all  values  of  T  in  the 
interval  O^T  <!  T,  where  T  is  finite,  but  otherwise  arbitrary. 
The  condition  for  periodicity  is  simply 

p'  =  OatT  =  7\  (11) 

If  we  choose  T  —  w,  an  inspection  of  equation  (10)  shows  that  for  e  =  Q  the 
solution  for  p  is  periodic  with  the  period  2ir,  whatever  may  be  the  value  of  8. 
Consequently  equation  (11)  must  carry  e  as  a  factor.  After  integrating 
equation  (10),  we  find  that  the  condition  (11)  is,  explicitly, 

0=  -[f+ajirte-  [f  +  fa.  +  ^+fojTr^+higher  degree  terms.      (12) 

Upon  dividing  out  the  factor  e,  there  remains  an  equation  in  which  the 
linear  terms  in  8  and  e  are  present,  and  this  equation  can  be  solved  for  8  as 
a  power  series  in  e.  We  find 


5=     -1+201  M2+       0:-0;X  +  •••«  +  (13) 

If  this  value  of  8  be  substituted  in  equation  (10),  it  will  then  admit  periodic 
solutions  for  p  having  the  period  2?r  for  all  values  of  e  sufficiently  small. 
Furthermore  the  solution  as  a  power  series  in  e  is  unique. 

64.  Existence  of  Periodic  Orbits  which  are  Inclined  to  the  Equatorial 
Plane.  —  For  p,2  =  0  the  differential  equations  (6)  admit  the  circular  solution 

c2^!,          r=l,          r  =  r,          7  =  0.  (14) 

In  order  to  investigate  the  existence  of  orbits  not  lying  in  the  equatorial 
plane,  but  having  the  period  2  IT  for  (j,27*Q,  let  us  put 


r=l+p,  q  =  0  +  <r,  c'=l  +  e,  (15) 

and  take  the  initial  conditions 

P  =  a,         p'  =  0,          (7  =  0,         ff'  =  /3/j:. 
The  conditions  for  periodicity  are  then 

p'  =  <r  =  0     at  T  =  TT. 

We  have  three  arbitrary  constants  at  our  disposal,  a,  0,  and  e,  and  two 
conditions  to  be  satisfied.  We  will  therefore  let  /8  remain  arbitrary  and 
determine  a  and  e  so  as  to  satisfy  the  two  conditions. 


ORBITs     UJnl    |      s\    DHI.VTK    M'MKIOHD.  1(1.") 

After  making  the  substitution-    l."()  and  expanding,  equations  (6)  become 

+  4p0J/**  +  higher  degree  term-.  (jg) 

!>  <r"+ff  =  3pff  —  6p*  ff  +  -<r*  —  ff  <%(**+  higher  degree  terms. 

In  order  to  integrate  these  equations  let  us  put 

p=(  2_op(,tt'aV.  <r=(  S^c'aV-  (17) 

The  pllt  and  ffllt  can  be  found  by  successive  integrations,  the  constants  of 
integration  being  determined  so  as  to  satisfy  the  initial  conditions.  In  the 
series  thus  obtained  put  T  =  TT.  The  two  conditions  for  periodicity  give 
the  two  equations 


is, 


Equation  18<i)  involves  only  the  even  powers  of  n,  while  (186)  involves 
only  the  odd  powers.  After  dividing  (186)  by  fa,  we  can  solve  it  fort  as  a 
power  >ei  ies  in  a  and  n'  of  the  form 


"  p'(T)=0  =  «, 

(6)         ff  (7r)  =0  =  ^(6,6+6, e'+^a'+^aHAMM-   '   '  '  ), 


t  =  r  ,  a'+f,  M'+f,  a'+c,  aM*+c,  M4+  (19) 

On  substituting  (19)  in  (18a),  we  obtain  a  series  of  the  form 
(a)  0  =  J1aMl+rf!a1+rfjM4+rf4aVJ+rfJa4+   •  •  •   •  (20) 

If  in  this  equation  we  make  the  substitution 


we  obtain 


which  can  be  solved  uniquely  for  y  as  a  power  series  in  p.  This  solution 
substituted  in  (206)  gives  a  as  a  power  series  in  n*.  This  value  of  a  sub- 
stituted in  (19)  gives  t  as  a  power  sei  ie-  in  p.  We  thus  have  a  solution 


where  P,  and  P,  are  power  series  in  ft1.  Newton's  parallelogram  shows  that 
equation  (20a)  has  two  additional  solutions,  but  as  they  are  imaginary  w 
we  shall  not  develop  them. 


106  PERIODIC    ORBITS. 

65.  Existence  of  Orbits  in  a  Meridian  Plane. — If  in  equations  (6)  we  put 
the  area  constant  c  equal  to  zero,  the  motion  of  the  particle  is  in  a  meridian 
plane;  that  is,  the  plane  has  ceased  to  revolve,  and  the  orbit  in  this  plane  is 
the  true  orbit.  After  changing  to  polar  coordinates  by  the  substitution 

r  =  p  cos<p,  <1  =  P  sin^>, 

the  differential  equations  are 


-  +  |  cos2<p+  j 


\  sin2<p  —  \  sin  4^5 


For  M2  =  0,  equations  (21)  have  the  periodic  solution 


(21) 


that  is,  a  circle.     For  M2^0,  we  will  put 

P=l+P, 

with  the  initial  values 

p  =  a,     p'  =  0,     a  =  0,     ff'  =  p, 

where  a  and  0  are  two  new  arbitraries.  By  §§  14-16,  p,  p',  a,  and  a'  are 
expansible  as  power  series  in  a,  /3,  and  /r2  with  r  entering  the  coefficients. 
The  conditions  for  periodicity  are  that  at  T  =  TT 

P'  =  <T  =  Q. 

If  we  integrate  equations  (21)  and  then  put  T  =  TT,  we  obtain  from  the 
periodicity  conditions  two  equations  of  the  form 

(a)  <T(7r)=0  =  a1 

(b)  p'(T)=0= 

The  first  of  equations  (22)  can  be  solved  for  a  as  a  power  series  in  /8 
and  M2-  This  expression  for  a  substituted  in  (6)  gives  rise  to  an  equation 
of  the  form 

(c)  0  =  Cl/V+c2/33+c30V+c4M4+ 

This  equation  has  the  same  form  as  (20)  and  can  be  solved  in  the  same 
way,  giving  a  solution  for  /3  as  a  power  series  in  ^2,  vanishing  with  /t2.  This 
expression  for  0  substituted  in  the  equation  for  a  gives  a  unique  value  for 
a  as  a  power  series  in  if,  vanishing  with  ju2.  Therefore  periodic  orbits  exist 
for  M2^0,  which  are  analytic  continuations  of  circular  orbits  for  /t  =  0. 

We  have  thus  proved  the  existence  of  the  following  three  classes  of 
periodic  orbits  which  have  the  period  2-n". 

I.  Orbits  lying  in  the  equatorial  plane  whose  generating  orbit  is  a  circle. 
II.  Orbits  inclined  to  the  equatorial  plane  whose  generating  orbit  is  a  circle. 
III.  Orbits  in  a  meridian  plane  whose  generating  orbit  is  a  circle. 


M|(H1I>     A  H«il    I      AN     M|t|.  Ml.    -l'IH.K(ill).  1(1? 


66.  Construction  of  Periodic  Solutions  in  the  Equatorial  Plane.     \\  e 

consider  first  orbits  in  the  r<|ii;ttunal  plane.  We  take  the  differential  (-(illa- 
tions (8),  and  by  means  of  the  transformations  there  given  we  proceed  at 
once  to  the  integration  of  equation  (10).  It  was  shown  in  equation  (13) 
that  6  can  be  expanded  uniquely  as  a  power  M-ries  in  c  in  such  a  manner 
that  the  solution  for  p  a.s  a  power  series  in  <•  shall  he  periodic  with  the 
period  'Iv.  Since  the  series  is  j)eriodic  with  the  same  period  for  all  values 
of  (  Miflicieiitly  small,  it  follows  that  the  coefficient  of  each  power  of  e  is 
it  -elf  periodic.  Since  the  solution  exists  and  is  unique,  it  must  be  possible 
to  determine  the  6  uniquely  by  the  condition  that  the  solution  shall  be 
periodic.  In  the  existence  proof  it  was  shown  that  6  vanishes  with  e. 
Therefore  p  and  6  have  the  form 

P  =  P0+P1e+p,e5+p,e'+  •  •  •  ,        «-*,  e+61e1-|-«Je>+  •  •  •  •         (23) 


The  p  are  to  be  determined  by  the  integration  of  equation  (10)  and  by  the 
initial  vah, 


P(0)=-l,  -0.  (24) 

The  5,  are  to  be  determined  in  such  a  manner  that  the  p,  shall  be  periodic. 
Upon  substituting  (23)  in  (10)  and  equating  the  coefficients,  we  find 


(6) 
(c) 


(25) 


These  equations  can  be  integrated  in  succession.     The  solution  of  (a)  which 
satisfies  the  initial  conditions  is 

p0=-cosT.  2(1. 

Since  the  initial  conditions  are  independent  of  c,  every  p,  except  p0  must 
vanish  at  T  =  0.      On  substituting  (26)  in  (256)  and  integrating,  we  have 

pl  =  $1(l-cosT)  +  [3-30y-^+6$Ml+        -][|—|co8T-|cos2T]-  (27) 

The  constants  of  integration  in  equation  (27)  have  been  determined  so  as 
to  satisfy  the  initial  conditions,  but  the  constant  5,  is,  as  yet,  undetermined. 


108  PERIODIC    ORBITS. 

On  substituting  the  values  of  p0  and  p^  in  (25  c),  we  find 


-)] 


COST 


2  •  •   -)]cos2T 

+  [3-402M2  •  •  -]cos3T. 

In  order  that  the  solution  of  this  equation  shall  be  periodic  the  coefficient 
of  cos  T  must  be  zero.  This  is  the  condition  that  determines  Sl  ,  and  con- 
sequently 


(28) 


which  agrees  with  (13)  of  the  existence  proof.  With  this  value  of  dl  equa- 
tion (28)  becomes 

f^r+P2  =  [S2+302M2  •  •  -]-H-902M2  •  •  •]cos2T+[3-402y  .     .]cos3T. 
The  solution  of  this  equation  which  satisfies  the  initial  conditions  is 

-f02M2  +   •  •  -JcosT 
+  |02M2+          -]cos3T. 

The  constant  52  is  as  yet  entirely  arbitrary.  It  is  determined  by  the 
periodicity  condition  on  p3  in  the  same  manner  that  6,  was  determined  by  the 
periodicity  condition  on  p2  .  Without  giving  the  details  of  the  computation, 
its  value  is  found  to  be 

52=-602M2  + 

This  method  of  integration  can  be  carried  as  far  as  is  desired.  In  order  to 
show  this,  let  us  suppose  that  p0  ,  .  .  .  ,  pn_,  have  been  computed  and  that 
all  the  constants  are  known  except  5n_!  .  From  (25  d)  we  have 

•]2PoPn_1+/n(po,  •  •  •  ,  P.-,),        (30) 


where  /„  (p0  ,  .  .  .  ,  pB_2)  is  a  polynomial  in  the  ps  and  contains  only  known 
terms.     It  is  easy  to  see  that  pn-\  depends  upon  5n_t  in  the  following  way, 

pn_i  =  5n_i(l  —  cosl)+  known  terms. 
Equation  (30)  may  therefore  be  written 


2//  .  .  .]«._,  COST 
+  [3-302M2+  •  •  •]  cos2T+  known  terms. 


ORBITS    AHot'T    AN    OBLA  I  K    >1>HKI«UD.  109 

In  order  that  the  solution  of  this  equation  shall  he  period  ic  the  coefficient  of 
CO>T  mu-t  he  /ero.  This  condition  determines  $„_,.  The  e(|uation  can 
then  he  integrated,  and  the  constant>  of  integration  will  he  determined  by 
the  conditions  that,  at  T  =0, 

dp, 
P*  =  -r=.  =0. 


Kveryth'mn  is  then  determined  with  the  exception  of  6,,  and  we  have 
p.=  (1  —  cosT)5,+  kimwn  terms. 


On  substituting  the  values  of  5,  and  5,  in  the  solution  as  far  as  it  has 

been  computed,  we  find 

p0=  —COST, 

COST 


[f  -i*lM*+  •  •  -JcosT   +   [30V-     -]cos2T 


From  these  expressions  the  series  for  r  becomes 

(a)     r=l-ecosT  +  j[f  +  {0y+(0t-4^)M4+  •  • 

•  •  -JcosT 
4  •  •  •]cos2T]e'  (31) 


•]cos3T  }(?+ 


On  suhstituting  this  value  of  r  in  the  equation  (86),  transforming  to  the 
independent  variable  T,  and  integrating,  we  find 

(6)    t,-r0= 


-JsinTJ 


110  PERIODIC    ORBITS. 

Equations  (31a)  and  (31&)  are  the  periodic  solutions  sought.  If  we  return 
to  the  symbols  denned  in  the  original  differential  equations  (1)  by  means  of 
equations  (5),  with  the  additional  notation 


n  VT-0?M2-302M4  •  '  '  =  ", 
we  have  the  following  expressions  for  the  polar  coordinates  : 


(32) 


v  —  v0  = 


(f 


(33) 


Equations  (32)  and  (33)  contain  four  arbitrary  constants,*  a,  e,  va,  and  £„. 
Since  the  differential  equations  of  motion  in  the  equatorial  plane  are  of  the 
fourth  order,  these  series,  within  the  realm  of  their  convergence,  represent 
the  general  solution.  The  expression  for  the  radius  vector,  R,  is  always 
periodic  with  the  period  2-ir/v.  At  the  expiration  of  this  period  v  has 
increased  by  the  quantity 


in  excess  of  2?r;  that  is,  the  line  of  apsides  has  rotated  forward  by  this 
amount.  If  6  is  commensurable  with  unity  the  orbit  is  eventually  closed 
geometrically.  If  6  =  I/J  ,  where  /  and  J  are  relatively  prime  integers, 
then  v  =  2(I  +  J)ir  at  t  =  2Jir/v,  and  the  particle  is  at  its  initial  position 
with  its  initial  components  of  velocity.  The  particle  has  completed  I+J 
revolutions,  and  the  line  of  apsides  has  completed  /  revolutions.  The 
mean  sidereal  period  is 

(35) 


Equation  (34)  for  the  rotation  of  the  line  of  apsides  has  an  interesting 
application  in  the  case  of  Jupiter's  fifth  satellite.  On  the  hypothesis  that 
Jupiter  is  a  homogeneous  spheroid  whose  equatorial  diameter  is  90,190 
miles  and  whose  polar  diameter  is  84,570  miles,  that  the  mean  distance  of 
the  satellite  is  112,500  miles,  that  the  eccentricity  of  its  orbit  is  .006,  and  that 

*The  constant  a  is  also  contained  implicitly  in  v  through  the  constant  n,  and  t  can  obviously  be 
replaced  by  (l  —  t,)  since  I  does  not  occur  explicitly  in  the  differential  equations  (1). 


ultHllS    AUtUT     \\    OUI.AIK    M'HKKOID.  Ill 

its  sidereal  period  is  llk  Om  22!?,  equation  :\\  gives  for  the  rotation  of  the 
lino  of  apside-  1  1  liT  per  year.  Thr  values  derived  from  observations  arc 
somewhat  discordant,  hut  are  in  the  neighborhood  of  883°  per  year.  If  we 
still  keep  the  hypothesis  that  Jupiter  is  homogeneous  in  density  and  of  the 
same  oblateness  as  before,  we  areconipelled  to  suppose  that  the  value  adopted 
for  its  polar  radius  was  about  9,000  miles  too  great.  In  reality  Jupiter  must 
be  much  more  dense  at  the  center  than  at  its  surface,  and  therefore  it  is  not 
necessary  to  suppose  so  large  a  reduction  in  making  an  allowance  for  its 
atmosphere. 

67.  Construction   of  Periodic   Solutions   for   Orbits   Inclined   to  the 
Equatorial  Plane.— By  means  of  the  area  integral  the  problem  has  been 

reduced  to  the  three  equations  (6),  the  first  two  of  which  are 


(«)  >*=r 

(6)  g*=- 

After  the  solution  of  these  equations  has  been  obtained,  the  third  coordinate 
is  found  from  the  equation 

(c)  "'=F'' 

\Ve  have  already  proved,  equations  (14)  to  (20),  the  existence  of  periodic 
solutions  of  these  equations  of  the  following  type: 

r  =     I+PJ  M*+p4M4+  •  •  •  ,  (36) 

q  =g,  M+g»Mr 
with  the  initial  conditions 

r'(0)=g(0)=0, 

the  constant  /S  being  arbitrary.  We  can  therefore  integrate  the  equations 
so  as  to  satisfy  these  initial  conditions,  and  determine  the  c,  in  such  a  manner 
as  to  render  the  solution  periodic. 

On  substituting  (36)  in  (6)  there  results 

<7i  a, 4-6  p,  a? 

(37) 


(38) 


Equation  (37)  contains  only  the  even  powers  of  n,  and  (38)  only  the 
odd  powers.     For  the  integration  we  have: 


f  ri+stfgjM'+IX+g.-Sftg,-  f 


112  PERIODIC    ORBITS. 

Coefficient  of  p.     The  coefficient  of  fj,  is  defined  by  q'i+qi  =  0,  and  the 
solution  of  this  equation  satisfying  the  initial  conditions  is 

^  =  /3sinr.  (39) 

Coefficient  of  p.   The  coefficient  of  //,  from  (37),  is  denned  by 


The  solution  of  this  equation  which  satisfies  the  assigned  initial  conditions  is 


The  constant  c2  is  determined  by  the  periodicity  condition  on  q3,  where  it 
is  found  that  it  must  have  the  value  c2  =  20,  —  /32;  and  a2,  which  is  deter- 
mined by  the  periodicity  condition  on  p4  ,  is  found  to  be  zero.  If  we 
anticipate  these  determinations,  we  have 


Coefficient  of  p.3-    The  coefficient  of  (j?,  from  (38),  is  defined  by 
&+?3  =  ?.(3p2+ftf-302)  =  (302  +  3c2-602)£sinr  +  Jo2/3sin2r. 

In  order  that  the  solution  shall  be  periodic  it  is  necessary  that  the  coefficient 
of  sinr  be  zero.  Therefore  c2=26~l  —  p'2.  On  substituting  this  value  and 
integrating,  we  find 

&  =  0,  sin  r--~-a2/3  sin2r. 
From  the  initial  conditions  we  must  have  q'3(Q)  =0,  and  therefore 


But  it  will  be  shown  in  the  next  step  that  a2  =  0,  and  consequently  that 

ft  =  0.  (41) 

Coefficient  of  ^.    It  follows  from  (37)   that   the    coefficient   of  /x4   is 
defined  by 

.        (42) 


ORHITS     \H<>lT    AN    OBLA1K    H'HKKOlD.  113 

Before  expanding  the  right  member  of  this  equation  we  will  examine  the 
coefficient  of  cosr,  which  we  know  must  he  zero  from  the  periodicity 
condition.  It  is  noticed  in  the  first  place  that  terms  in  COST  can  arise 
only  through  terms  involving  p.  and  */,  ,  and  .secondly  that  all  such  terms 
carry  a,  as  a  factor.  No  other  arbitrary  enters  the  coefficient  ;  therefore  we 
must  take  o,  =  0.  It  can  be  shown  by  induction  that  the  arbitrary  con- 
stant a,(the  coefficient  of  COST),  which  arises  in  the  integration  of  p,  ,  is 
determined  1>\  the  periodicity  condition  on  p{Jrt  ,  and  further  that  its  value 
is  zero.  The  proof  is  omitted  for  the  sake  of  brevity. 

Upon  substituting  the  value  a,  =0  in  p,  and  qt  and  expanding  the  right 
member  of  (42),  we  find 


Since  the  constants  of  integration  must  both  be  zero,  the  solution  is 


1  1"  we  anticipate  the  value  of  c4  which  is  found  below,  we  have 


Coefficient  of  //.     We  find  from  (38)  that  the  coefficient  of  /**  is  defined  by 


From  the  periodicity  condition  we  have 


(  )n  integrating  and  imposing  the  initial  conditions,  we  find  at  this  step 

(43) 


This  is  sufficient  to  make  evident  the  general  character  of  the  series. 
The  r-equation  contains  only  even  multiples  of  T  and  the  ^-equation  con- 
tains only  odd  multiples.  The  r-equation  contains  only  even  powers  of  n 
and  of  T,  while  the  (/-equation  is  odd  in  both  these  respects.  The  series 
are  therefore  triply  even  and  odd. 


114  PERIODIC    ORBITS. 

On  collecting  the  various  coefficients,  we  have  the  following  series: 
(a)  r- 


(b) 

~ "  '  "  '  '  (44) 


64  r 

(d)  c2=l  +  [202-/32]M2+[-404.-^0*02]/u4+ 

In  this  solution  the  constants*  a,  /3,  va,  and  TQ  are  arbitrary.     As  is  shown 
by  equation  (44c)  the  nodes  regress,  the  measure  of  regression  being 


The  generating  orbit  of  these  solutions  is  a  circle  in  the  equatorial 
plane.     A  circle  having  any  assigned  inclination  might  have  been  used,  e.  g., 


r  =  Vl  —  s2  sin2T,  g  =  ssinr,  w  =  tan~1  [Vl  —  s2  tanr],       (45) 

where  s  is  the  sine  of  the  inclination.  The  solution  thus  obtained  would 
have  been  identical  with  (44) .  If  we  should  expand  (45)  as  power  series  in 
s  and  put  s=  /3/x,  we  should  find  that  the  terms  thus  obtained  are  identical 
with  the  terms  independent  of  6\  in  the  solution  which  has  been  worked  out. 
It  might  therefore  be  of  assistance  in  the  physical  interpretation  of  the 
constants  to  put  /3/*  =  s  in  the  series  (44). 

68.  Construction  of  Periodic  Solutions  in  a  Meridian  Plane. — When  the 
constant  c  is  zero  the  motion  is  in  a  meridian  plane.  The  equations  of 
motion  (21)  are 


(46) 


We  have  proved  in  §  65  the  existence  of  periodic  solutions  of  these  equations 
as  power  series  in  tf,  which,  for  M2  =  0,  reduce  to  the  circle  p  =  l,  v  =  r. 
Let  us  therefore  put 


*The  constant  a  is  contained  implicitly  through  r  and  9?;  see  equations  (4). 


i  "KBITS   ABOUT   AN   OBLATK    SI'HK1«HD.  115 

Upon  substituting  tlu-x-  e\i>n->iuns  in  (40),  expanding,  and  collecting  the 

eoeilieients  (if  the  vari">u>  powers  of  p,  we  find 


-2  sin2r  +  sin4r)  tf  p, 


(47) 


The  initial  e<>ii<Iitionsarcp'(0)  =^>(0)  =0.    On  proceeding  to  the  integration, 

we  have: 

Coefficients  of  /i1.     The  coefficients  of  M'  are  defined  by 
(a)  p*  —  3p,-2v>j  =  -  —  ^t 


(48) 
(b) 


On  integrating  (6)  once,  we  have 

(c)  ^=-2p,+^  cos  27- 

If  we  substitute  this  value  of  <p't  in  (a),  the  latter  becomes 

(d)  p,'+pt=2Cl+0f)-0Jcos2T-    0Jcos4r. 


The  integration  of  this  equation  gives  for  the  general  solution 

p,  =  (2c,+|0j)  +ctsinr+ct  cos  r+|0J  cos  2r+^  cos  4r. 

Since  p't  (0)  =  0  we  must  take  c,  =  0.     On  substituting  this  value  of  p,  in 
(c)  and  integrating,  we  get 


-  3c,-     0?r-2c,sinT-      0Jsin2T-      (?J  sin4r+c4. 

From  the  initial  conditions,  <pt  must  be  zero  when  r  =  0;  therefore  c4  =  0. 
It  must  also  be  periodic;  therefore  2c,=  —  dj.  All  of  the  constants  of 
integration  are  now  determined  except  c,  ,  which  will  be  determined  by  the 
periodicity  condition  on  p4  . 


116  PERIODIC    ORBITS. 

The  differential  equations  for  p4  and  <p4  are  just  the  same  as  for  p2  and  <?2 
except  in  the  right  members.  The  process  of  integration  is  just  the  same.  In 
the  right  members  only  even  multiples  of  T  occur  except  in  terms  carrying  the 
undetermined  constant  c3  as  a  factor.  In  the  equation  corresponding  to  (48d) 
there  will  be  a  term  in  COST  carrying  c3  as  a  factor.  But  the  integral  from  this 
term  will  be  non-periodic  unless  cs  =  0.  Upon  putting  c3  =  0,  the  integration 
proceeds  just  as  before  and  the  constants  are  determined  in  the  same  manner. 
The  same  steps  are  repeated  in  the  coefficients  of  //,  and  so  on  for  all  higher 
powers.  Therefore  no  odd  multiples  of  T  occur  in  the  solution.  We  have, 
therefore, 


(49) 


Since  the  series  involve  only  even  multiples  of  T,  the  orbits  are  symmetrical 
with  respect  to  both  the  r-axis  and  the  g-axis. 

This  completes  the  formal  construction  of  the  solutions  of  which  the 
existence  was  proved  in  §§63,  64,  and  65. 

II.    ORBITS  RE-ENTRANT  AFTER  MANY  REVOLUTIONS. 

69.  The  Differential  Equations.  —  The  orbits  which  we  have  previously 
considered  have  had  the  common  property  of  involving  only  the  period  2ir. 
Since  this  period  is  independent  of  the  oblateness  of  the  spheroid,  the  deriva- 
tion of  these  orbits  has  been  relatively  simple.  We  shall  proceed  now  to 
investigate  a  class  of  orbits  which  involves,  beside  the  period  2ir,  another 
period  2T/X,  where  X  is  a  function  of  the  oblateness  of  the  spheroid,  the 
inclination  of  the  orbit  to  the  equator,  and  the  mean  distance  of  the  particle. 
We  will  start  out  from  the  solution  which  involved  an  arbitrary  inclination 
(§67).  Into  this  solution  four  arbitrary  constants  were  introduced,  viz., 
inclination,  mean  distance,  longitude  of  node,  and  the  epoch.  Two  more 
arbitraries  are  necessary  for  a  complete  solution,  viz.,  constants  correspond- 
ing to  the  eccentricity  and  to  the  longitude  of  perihelion.  In  what  follows 
we  shall  introduce  the  constant  corresponding  to  eccentricity. 

We  have  found  for  the  differential  equations  a  certain  solution,  given 
in  (44),  which  we  may  write 

-  = 


which  is  symmetric  with  respect  to  the  equatorial  plane.  That  is  to  say,  at 
T  =  0  the  particle  is  in  the  equatorial  plane  and  its  motion  is  perpendicular 
to  the  radius  vector.  Its  initial  distance  is  <p  (0). 

Suppose  now  we  change  the  initial  distance  slightly  and  also  the  initial 
velocity  so  that 

,         g(0)=0,         r'(0)=0,         ?'(0)  =*'(())  +7, 


ORBITS  ABOUT   AN   OBLATE   SPHEROID. 


117 


and  irive  an  increment  to  the  constant  of  areas  so  that  c*  =  rJ+«.  Can 
we  determine  the-e  three  constants  a,  -y,  and  «,  as  functions  of  /3  and  n,  in 
such  a  manner  t  hut  the  scries  for  r  and  q  shall  be  periodic?  The  solutions  can 
be  expressed  in  the  form 


where  p  and  a  are  the  necessary  additions  to  <p  and  ^.  If  now  we  substitute 
these  expressions  in  the  differential  equations  (6),  all  the  terms  independent 
of  p,  <r,  and  t  will  drop  out,  and  there  will  remain  the  following  differential 
equations  for  p  and  <r: 


(a) 


jp1- 


[-12/3 


po- 


(6) 


(50) 


[f 


In  the  first  of  these  equations  the  coefficients  of  all  the  terms  containing 
odd  powers  of  a  involve  only  sines  of  odd  multiples  of  r  and  odd  powers  of 
M;  all  other  coefficients  involve  only  even  powers  of  M  and  cosines  of  even 


118  PERIODIC   ORBITS. 

multiples  of  r.  In  the  second  equation  the  coefficient  of  every  odd  power 
of  a  involves  only  even  powers  of  ^  and  cosines  of  even  multiples  of  r;  all 
other  coefficients  involve  only  odd  powers  of  /*  and  sines  of  odd  multiples  of  T. 
These  properties  play  an  important  role  throughout  the  entire  discussion. 

70.  The  Equations  of  Variation.  —  Considering  merely  the  terms  of  the 
differential  equations  (50)  which  are  linear  in  p  and  a,  we  have 


(a) 


sinr-f /33sin3T~U3+  •• 

o  J 

r       r     o         3  -,  1         IY  n 

(6)     <r"+-U  +   --fl2+-/32  cos2r  L2+  '  '  '  r°"+l    -3/3smr  hi 
LL22  J  JLL  J 

T 

+  [(~302^+|/33jsinT  -  -^3  sin3T]/Li3+  •  •  4p  =  0. 
The  solutions  of  these  equations  have  the  general  form 

p=  S  c^/Vj(T)>  <r  =  S  C/fr*V/(f)f 


(51) 


where  c,  and  X.,  are  constants,  and  <P}(T)  and  $,(T)  are  periodic  functions  of  r 
with  the  period  2ir.  The  four  values  of  the  X,  (real  or  imaginary)  are 
associated  in  pairs,  equal  in  value  but  of  opposite  sign  (§33);  and  since  the 
solution  (44)  contains  two  arbitrary  constants,  t.  e.,  the  origin  of  time,  TO, 
and  the  mean  distance,  a,  it  is  known  a  priori  that  one  pair  of  the  X^  has 
the  value  0  (§33).  If  we  suppose  that  A3  =  X4  =  0,  the  two  corresponding 
solutions  have  the  form 


The  values  of  <p3(r)  and  ^3(r)  can  be  obtained  at  once  by  differentiating 
the  solutions  for  r  and  q  [equations  (44)]  with  respect  to  T;  and  the  values 
of  [<p4(r)+T  <f>3M]  and  [^W+r&M]  are  obtained  by  differentiating  ar 
and  aq  with  respect  to  a.  Thus  two  of  the  solutions  of  the  fundamental  set 
can  be  found  without  integration. 

We  will  consider  first  the  two  solutions  in  which  the  X,  are  not  zero. 
Let  us  substitute  in  (51)  the  expressions 


ORBITS  ABOUT  AN   OBLATE   SPHEROID.  119 

After  dividing  out  the  exponential,  there  remains 
(a)     * 


,  f>2 1 

•        V  I  i~*  *«  I  /  I  r        «  1  *  I  1  •  I  t  1  .  1  .  .  r  I  ••  1  ft: 

(o) 


where 

a,=  —  3/3  sin  T, 


64  =  a  sum  of  cosines  of  even  multiples  of  T. 

With  respect  to  equations  (52),  it  is  known  that  <p  and  ^  are  periodic  with 
the  period  2jr  and  that  X  vanishes  with  n,  since  the  problem  then  reduces 
t<»  the  two-body  problem,  in  which  the  characteristic  exponents  are  all  zero. 
Since  <p,  \f/,  and  X  are  analytic  in  n,  we  may  put 


l-a  i-o  i-o 

The  expressions  for  <p,,tj,  and  X  are  determined  from  (52)  as  follows  : 
Coefficients  of  /*"•   The  terms  independent  of  n  are  found  to  be 


™ 


0,  *.  =  7?"  COST  +  7™  sinr. 

Coefficients  of  p.    The  differential  equations  for  the  terms  in  n  are 

*f+*E--2t'XHPb-ai*.i  ^'+^,  =  -2tX1^-al^,.          (54) 

Since  the  periodicity  conditions  demand  that  the  coefficients  of  COST  and 
sin  T  in  the  right  members  shall  be  zero,  we  must  take  X,  =  0,  after  which  we 
get,  on  making  use  of  (53), 


cos2T  +    0Ti0)  sin2r, 


(55) 


Upon  integrating,  we  have 

*  =  <»  COST  +  oi"  sinT  +  f  j9  7,""  +  1/3  7i0>  cos2T  -\fiyT  sin2T, 

^a0)-|-a0>  cos2T-    ^  C  sin2T. 


(56) 


120  PERIODIC   ORBITS. 

Coefficients  of  tf.    The  coefficients  of  //  are  defined  by 
*£+ft  =  -2tX2<po-a2<A>-ai'/'i  ,        if'*  +&  =  -2i\t'0-b^0-a1<p1  ;     (57) 
or,  expanding  by  (53)  and  (56), 

ft  +  ft  =  f  0  Ti"  -  |/3  7"'  cos2r  +  f  0  7!"  Sin2r  +  [40°  af  +  2iX,oH  sinr 

2ar  cos3r, 


-f/Sft?1  sin2T  +  |_y182720)+2iX27|0)J  sinr 
+2tX2720>  COST. 

In  order  to  satisfy  the  periodicity  conditions  we  must  have 

2  I  (59) 

40i  a<0>  -  2iX2a20)  =  0,  +2t'X2720)  =  0. 

The  last  two  of  these  equations  are  satisfied  by  taking 
On  solving  the  first  two,  we  find 


Equations  (59)  can  also  be  satisfied  by 

X2  =  a<°>  =  af  =  yf  =  0,  T[O)  arbitrary. 

This  leads  to  the  development  of  the  solution  in  which  the  characteristic 
exponent  is  zero,  which  will  be  discussed,  beginning  with  equations  (80),  by 
using  the  integral  relations. 

It  was  known  at  the  outset  that  the  two  values  of  X  are  equal  but  of 
opposite  sign.  We  will  choose  the  one  with  the  positive  sign.  The  solution 
for  the  negative  X  can  be  derived  from  it.  The  condition  ai0>  —  ia20)  =  0 
still  leaves  us  with  an  arbitrary  constant.  Since  the  equations  are  linear, 
this  constant  will  enter  the  solution  linearly,  and  may  therefore  be  taken 
equal  to  unity,  inasmuch  as  the  solution  after  development  is  multiplied 
by  an  arbitrary  constant.  We  will  take  then  a[0>  =  l,  which  therefore 
makes  o.20)  =  —  i.  Consequently 


—  i  sinr,  ^c  =  0-  (60) 

On  integrating  equations  (58)  with  these  values  of  ai0>  and  a20>,  we  get 


snr  + 


-3/32t  sin3r, 


(61) 


ORBITS   ABOUT   AN    OBLATE   SPHEROID.  121 

Coefficients  of  p*.     The  terms  of  the  third  decree  in  /*  are  defined  by 
+10y? 


in  2r-f  fty?  cos2r, 


(62) 


From  the  periodicity  conditions  we  must  have 


The  last  two  equations  can  be  satisfied  only  if  y"}  =  y(t"  =  0.  The  first 
two  can  be  satisfied  only  if  X,  =  (aj"  -  1  oj")  =  0.  The  condition  a',"  -  1  a<"  =  0 
again  leaves  us  an  arbitrary  constant;  it  gives  us  V>I  =  C(COST  —  i  sinr),  but 
this  is  the  same  as  <f>0  multiplied  by  c/x.  That  is,  the  solution  is  repeating 
itself  one  degree  higher  in  n,  and  this,  of  course,  should  be  expected  since  the 
equations  are  linear,  and  <f>a  multiplied  by  any  power  of  n  must  satisfy  them. 
We  may  then  choose  the  arbitrary  multiplier  equal  to  zero,  which  is  the  same 


as  choosing  a 


*», 


"'  =     "  = 


=  a 


=  0.     On  integrating  (62)  with  these  values,  we  find 


(63) 


It  can  be  shown  by  induction  at  this  point  that  <p  and  X  are  even  series 
in  n,  and  that  ^  is  an  odd  series  in  /x.  Furthermore,  v  contains  only  odd 
multiples  of  T,  and  ^  only  even  multiples.  Consequently 


and  all 


<vm  =  •v(2)  =  a(S)  =  a(S)  =  -v(1)  =  •v(S)  =  0 
Ti       Ti       "i       ai       Ti        it       ui 


Coefficient  of  /u4.     The  term  of  the  fourth  degree  in  n  is  defined  by 


snr 


cos  Sr+^t  / 

lo 


sin5r. 


(64) 


122  PERIODIC    ORBITS. 

The  conditions  that  <p4  shall  be  periodic  are 

-2   X4+402(a<2)-ia<2))-16    0i-10O2  =  0, 
+2i  X4+4  02  (t  a?>  +  af)  +  16  i  ^  =  0. 

On  solving  these  equations,  we  find 


In  the  last  equation  we  can  choose  af*  =  5/4  /32  and  e42)  =  0.  This  choice 
will  make  the  coefficient  of  sinr  in  <p2  equal  to  zero,  and  since  the  same 
thing  occurs  for  each  <p}  it  is  evident  that  the  corresponding  choice  will 
simplify  the  solution  by  making  the  coefficient  of  sin  T  equal  to  zero  for  all 
powers  of  /x-  We  have,  then,  on  integrating  and  collecting  results 


|/32cos3r+f 


<f>0  =  cos  T  —  i  sin  r, 


(65) 


The  coefficient,  a4  ,  of  COST  in  <pt  is  determined  by  the  periodicity  condition 
of  <pt.  That  this  process  of  determining  the  values  of  the  X,  and  the  constants 
of  integration  arising  at  each  step  is  general  can  be  shown  as  follows:  Let 
us  suppose  that  we  have  computed  all  terms  of  <p  up  to  and  including  <p, 
with  the  exception  of  the  constants  of  integration  in  <p}.  We  have  then 

<PJ  =  a"}cos  T  +  a^'sin  T  +  known  terms. 

It  follows  from  equations  (52)  that  the  a"'  and  a"'  enter  \f/i+l  only  as  shown 
explicitly  in 


Consequently,  in  so  far  as  if/J+l  depends  upon  these  terms,  it  is 


ORBITS   ABOUT   AN   OBLATE   SPHEROID.  123 

Similarly,  in  so  far  as  <f>j+.  depends  upon  constants  as  yet  undetermined,  it 
is  found  from  (-([nations  (52)  tliat 

f  |8I)+  f  / 


(66) 


where  A^,  and  B,+t  are  the  known  terms  in  the  coefficients  of  COST  and 
sinr  respectively.     From  the  periodicity  condition  we  must  have 


COST 


i=0.    (67) 
The  solution  of  those  equations  is 

.         (68) 


As  has  already  been  pointed  out,  we  can  choose  a"'  =  0,  and  we  have  then 


In  order  to  show  that  XH2  and  a{n  are  real,  it  will  be  sufficient  to  show 
that  Ai+t  is  real  and  that  BJ+1  is  a  pure  imaginary.  This  is  readily  proved 
by  induction,  for,  up  to  j  =  4  inclusive,  we  have 

t+i  t+i  /-i  /-i 

ipj=  S  W,COSICT+I  S  n^sinKT,          tj  =  i  2  /«COSKT+  2  ^sinicr, 

K-l  *-l  *-0  «-0 

\\liere  m«,  n,  ,  /,,  and  gt  are  all  real.  From  the  form  of  the  differential 
equations  it  follows  at  once  that  the  same  forms  hold  for  j  =  5,  then  6,  and 
so  on.  That  is,  A}+t  is  real  while  5/+,  is  purely  imaginary. 

Furthermore,  it  is  to  be  noticed  that  At+t  and  Bl+t  do  not  contain  any 
terms  in  /3  independent  of  6\  ,  and  consequently  the  0J  ,  which  appears  in  the 
denominator  of  a\'\  will  divide  out.  This  is  proved  as  follows:  If  0J  be 
put  equal  to  zero  in  the  differential  equations,  then  equations  (52)  become 
the  equations  of  variation  of  a  circular  orbit  in  the  ordinary  two-body 
problem,  the  plane  of  the  circle  being  inclined  to  the  plane  of  reference  by 
an  angle  whose  sine  is  Pn  =  s.  Equations  (6)  can  then  be  written 


where  the  constant  c1  is  given  the  form  (1  -s*)(l  —  e1).     For  these  equations 
we  have  the  solution 

(1  -  e5)  Vl-jPain'tf-'ft)  (l-eO»ain(J-O)         ,-,  v 

-  - 

where 


124  PERIODIC    ORBITS. 

If  now  we  form  the  equations  of  variation  by  varying  r,  q,  and  e,  that  is 
by  putting 


where  rc  ,  q0  ,  and  e0  are  the  values  in  (71),  we  find 

„      dRn,dRfr,dR  »_dQn,dQ     ,dQ 

p~-'--frp+^q-ff--~te€'  -aFp+aT  "^ 

Three  solutions  of  equations  (72)  are  given  by 

(1)  (2)  (3) 

3r0  dr0  dr0 


dq0 


- 

' 


If  e0  9^  0  these  three  solutions  are  distinct.  The  case  in  which  we  are  interested 
is  that  for  which  eg  =  0,  but  then  these  three  solutions  are  not  distinct,  for 
the  first  two  coincide,  as  is  readily  seen  by  putting  e  =  0  in  (71).  Since  the 
equations  are  linear,  it  follows  that 


-f-  - 

'C  ' 


-  - 

drj'  'idn       dr'  'tdSl       dr> 

is  also  a  solution;  but,  since  it  vanishes  with  e0,  it  carries  e0  as  a  factor  which 
we  can  divide  out  and  absorb  in  the  arbitrary  constant  c4  .  For  e0  =  0  this 
solution  does  not  now  vanish,  and  it  is  moreover  distinct  from  the  first 
solution.  Thus  we  have  three  distinct  solutions  even  when  e0  =  0,  but  since 
dQ/de0=0  and  dR/de0  carries  e0  as  a  factor,  equations  (72)  pass  over  to  the 
equations  of  variation  of  a  circle  when  e0  =  0.  For  these  equations  we 
have  therefore  three  solutions  which  are  periodic  with  the  period  2  TT.  The 
fourth  solution  is  not  periodic  for  it  involves  a  term  of  the  form  T  times  a 
periodic  function. 

Let  us  return  now  to  the  solution  which  we  have  developed,  (65),  and 
consider  only  the  terms  which  belong  to  the  two-body  problem,  viz.,  the 
terms  which  are  independent  of  B\  .  This  solution  can  be  separated  into 
two  solutions,  one  of  which  is  real,  the  other  purely  imaginary.  The  real 
solution  is  the  third  one  of  (73),  and  the  purely  imaginary  solution  is  the 
second.  Since  both  of  these  solutions  are  certainly  periodic  with  the  period 
2ir,  it  follows  that  no  terms  in  /3  alone  can  occur  in  the  A}+t  and  Bj+z,  (67), 
because  the  presence  of  such  terms  would  give  rise  to  non-periodic  terms  in 
the  two-body  problem.  Hence  AJ+Z  and  BJ+2  carry  0*  as  a  factor  which  can 
be  divided  out  of  equation  (69).  Furthermore,  XJ+2  ,  equation  (68),  carries  6\ 
as  a  factor,  and  therefore  X  vanishes  with  the  oblateness  of  the  spheroid. 


where 


ORBITS   ABOUT   AN    OBLA  1  K    Sl'll  KU<  HI).  125 

The  solution  which  we  have  obtained  may  be  written 


(76) 


By  putting  e'Xr  =  cosXr+i  sinXr,  the  solution  takes  the  form 

,  1 


We  have  thus  one  solution  of  the  differential  equations.    A  second  solution 
can  be  derived  from  it  by  merely  changing  the  sign  of  i,  or 


By  adding  and  subtracting  these  two  solutions,  we  have  finally 


A  and  B  being  arbitrary  constants. 

As  above  developed,  there  is  a  certain  arbitrariness  in  these  solutions, 
owing  to  the  manner  in  which  the  constants  of  integration  were  determined. 
They  may  be  reduced  to  a  normal  form  by  multiplying  each  solution  by  the 
proper  series  in  ^  with  constant  coefficients.  By  this  process  we  can  make 

p(1)(0)+P(2>(0)  =  1,  a("(0)  -<rro(0)  =/3M.  (79) 

Since  [p(l)  —  p(2)]  and  [<rw+ffa)]  are  sine  series,  they  vanish  for  r  =  0. 
The  third  and  fourth  solutions  of  the  equations  of  variation  are 

(80) 

the  r  and  q  being  defined  by  equations  (44).  In  performing  the  differentia- 
tion in  this  last  solution  it  will  be  remembered  that  T  and  0J  are  functions 
of  a.  The  third  solution  also  can  be  normalized  by  giving  the  arbitrary 
constant  such  a  form  that 

P,(0)  =  0, 


126  PERIODIC    ORBITS. 

As  already  stated,  the  fourth  solution  is  non-periodic  and  has  the  form 


where  <f>4  and  ^^  are  periodic  functions  of  r  with  the  period  2w.     As  in  the 
previous  solutions,  this  can  be  normalized  so  that  at  r  =  0 


The  functions  <p4  and  <£4  are  also  easily  found  by  substituting  (81)  in  the 
equations  of  variation  and  solving  for  these  variables  (which  must  be  periodic)  . 
Upon  carrying  out  the  foregoing  operations,  we  find  the  following 
fundamental  set  of  solutions: 

p=+4Jcos(l-A)r  +  [-j/32cos(l-A)r  +  |/32  cos(l  +  A)r 


-  sin(l  -X)r+[-^-^     sin(l  -  A)r-      /32sin(l-{-A)T 


-|^2  +  |^4)  cos2r-i/34cos4r]M4+ 


(82) 


<r=  +A         /3  sinAr  -    /3sin  (2  -  A)  TM+      02|3sinAr 


i  |3cos(2-A)r]/i+[-^2^cosA 


-J02/3+f/33)  sinr]M3+ 


(83) 


ORBITS   ABOUT   AN   OBLATE   SPHEROID.  127 

71.   Special  Theorems   for  the  Non-Homogeneous  Equations.     The 

general  theorems,  proved  in  Chapter  I.  Section  IV,  on  (lie  character  of  the 
solutions  of  non-homogeneous  linear  differential  equal  ions  \\  il  h  periodic  coef- 
ficients. pre>uppo-e  merely  the  conditions  that  the  coefficients  are  periodic 
with  the  period  27r.  Additional  facts  with  regard  to  the  solutions  can  lieestab- 
li.vhed  when  additional  fact.-  are  specified  with  regard  to  the  coefficients  of 
the  differential  equations.  The  equations  of  variation,  (51),  may  be  written 


where  the  notation  with  respect  to  the  6's  has  the  following  significance: 
F.verv  subscripts  denote  functions  even  in  T,  and  odd  subscripts  denote 
functions  odd  in  T;  one  dash  indicates  that  only  odd  multiples  of  T  are 
involved,  and  two  dashes  indicate  that  only  even  multiples  of  T  are  involved. 
The  solution  of  equations  (82)  and  (83)  may  be  characterized  in  the  same 
manner,  and  are  then 


(85) 


where  the  notation  is  the  same  as  for  the  0's  with  the  exception  that  in  the 
first  two  solutions  every  integral  multiple  of  T  is  increased  by  ±  XT,  e.  g., 
cos  (3+A)T.     On  these  terms  the  dashes  refer  only  to  the  integral  part. 
Suppose  now  we  have  the  non-homogeneous  differential  equations 


(86) 


where  g(r)  and/(r)  are  periodic  with  the  period  2r.     Since  the  character- 
istic exponents  are  0,  0,  =•=  V^l\,  by  §31,  the  general  solution  has  the  form 


(87) 


128  PERIODIC    ORBITS. 

where  the  (pt)  and  (o-,)  are  the  complementary  functions,  and  the  £,  and  rjt 
are  the  particular  integrals  of  which  the  ut  are  the  periodic  parts,  and 
where  a  and  b  are  constants  which  depend  upon  the  differential  equations. 
Let  us  suppose  that  g(r)  is  an  even  function  of  T  and  that/(r)  is  an  odd 
function  of  T,  and  let  us  seek  the  character  of  the  solutions  which  satisfy 
the  initial  conditions 


On  changing  r  into  —  T  in  equations  (86)  ,  we  get 


(88) 


From  equations  (86)  and  (88)  we  obtain,  by  eliminating  g  (T)  and/(r),  the 
differential  equations 

j-[Pl(T)—  Pi(  —  T)]=         [P2(T)+P2(  —  T)], 


(89) 


These  equations  are  the  same  as  the  original  homogeneous  set  (84) .    Hence 
their  general  solutions  have  the  same  form,  viz., 


r)  =  A  T,  (T)  +  B  ^  (r)  +  C  Tt  (T)  +£>[«;  (r 


(90) 


Upon  putting  r  =  0,  we  find  from  the  first  and  the  fourth  of  these 
equations  that 


ORIUI-     \UniT    AN    OBLATK    s|'MKI«  HI). 


129 


Hither  A  =  D  =  0.  or  the  determinant  i^(0)J^(0)-~«t(0)(0)«Q.  But  it  IB 
readily  verified  that  this  determinant  is  not  zero.  Therefore  A=*D  =  Q. 
By  virtue  of  the  hypothec-  made  on  the  initial  values,  it  follows  from  the 
second  and  third  equations  that 


and  hence  B  =  C  =  0;  consequently 

Pi(r)-p,(-r)=0, 


P,(r)+p,(-r)=0, 


Since  tliese  equations  are  identities  in  r,  we  have  the  following  theorem  : 

Theorem  I.  If  g(r)  is  an  even  function  of  T  and  f(r)  is  an  odd  function 
of  T,  and  if  p,(0)  =<r,(0)  =  0,  then  P,(T)  and  <T,(T)  are  even  functions  of  T,  and 
p,(r)  and  cr,(r)  are  odd  functions  of  T. 

I  n  t  he  same  way  it  can  be  shown  that  if  g(r)  is  odd  and  /(T)  is  even  and 
if  Pi<0)  =  a,(0)  =0,  then  p,  and  <r,  are  odd,  and  p,  and  <r,  are  even. 

Let  us  suppose  now  that  g(r)  contains  only  even  multiples  of  T,  and 
that  /(T)  contains  only  odd  multiples  of  T.  The  general  form  of  the  solution 
will  be  the  same  as  (87),  and  £,  ,  £,  ,  77,,  and  T;,  satisfy  the  differential  equa- 


(92) 


by  7j'(r).    Then  by  changing 


'.»:; 


Let  us  denote  £,(r+'r)  by  £,'(T)  and 
into  T+JT  in  (92),  we  have 


From  equations  (92)  and  (93)  it  follows  that 


!M 


130  PERIODIC   ORBITS. 

The  solutions  of  these  equations,  which  have  the  same  form  as  (84),  are 


(95) 


On  forming  these  expressions  directly  from  (87),  we  get 

3  —  &7r(ra3+a4), 


(96) 


A  comparison  of  equations  (95)  and  (96)  shows  that 

D=-bTr, 


W!(T)  —  WI(T+TT)=O,  a>3(T)+a>3(T+7r)  =0, 

a>2(r)-co2(r+7r)  =  0,  co4(r)+w4(r+7r)  =0. 


Therefore  c^  (r)  and  o>2  (T)  contain  only  even  multiples  of  T,  while  w3  (T)  and 
w4(r)  contain  only  odd  multiples  of  T,  and  by  carrying  this  result  into  equa- 
tion (87),  we  have 


-|T2  a3+r  a4],     %  =  c 


(97) 


These  results  may  be  expressed  in 


Theorem  II.  If  g(r)  contains  only  even  multiples  of  T  andf(r}  contains 
only  odd  multiples  of  T,  then  &  and  £2  contain  only  even  multiples  of  T,  and 
iji  and  ij2  contain  only  odd  multiples  of  T. 

If  in  addition  to  the  above  hypotheses  we  suppose  that  g(r)  is  an  even 
function  of  r  and/(r)  is  an  odd  function  of  T,  then  £t  and  r?2  are  even  functions 
and  &  and  ^  are  odd  functions;  therefore  6  =  0.  But  if  Q(T)  is  an  odd 
function  and/(r)  is  an  even  function  of  T,  then  &  and  ij2  are  odd  functions  and 
£s  and  ^  are  even  functions,  and  in  this  case  a  =  0. 


ciKHII-     \H(H    1     AN    OBLATK    SI'HKKOID.  131 

In  the  sunn-  manner  we  can  prove 

Theorem  I II.  //  g(r)  contains  only  odd  multiples  of  r  and  f(r)  contains 
unly  even  multiplf*  »f  T.  tli>n  £,  and  £,  contain  only  odd  multiples  of  T,  and  t?,  and 
»j,  contain  only  <  re//  multiples  of  T.  Furthermore  £ , ,  £, ,  17, ,  and  ijt  are  periodic 
with  th  i><  riod  2w. 

If  g(r)  is  of  the  form  £  m>cosO'±X)r  and  also  if  f(r)  has  the  form 
-\n(j±\)T,  then,  since  i-v/^X  arc  th<>  characteristic  exponents  of  the 
homogeneous  equations,  the  form  of  the  solution  is,  by  §§30  and  31, 


('.IS  | 


luit,  since  g(r)  is  an  even  function  and/(r)  is  an  odd  function,  £,  and  ij,arc 
t-vcn  functions,  and  £,  and  r;,  are  odd  functions.  Therefore  all  the  coefficients 
in  (98)  which  have  the  superfix  (2)  are  zero.  But  if  g(r)  were  an  odd 
function  of  T  and/(r)  an  even  function,  then  all  the  coefficients  in  (98)  which 
h:i\c  the  superfix  (1)  would  be  zero.  Therefore 

Theorem  IV.  //  g(r)  has  the  form  2mycosO'±X)T,  and  if  /(T)  has  the 
form  2n;  *in(.;'±X)r,  where  ±  V—  IX  are  the  characteristic  exponents  of  the 
homogeneous  equation,  then  the  particular  solution  has  the  form 


ij,  =  2p,  sin  (c±X) 


(00) 


From  similar  reasoning  we  have 


Theorem  V.  //  g(r)  has  the  form  2w,st'nO'±X)r  and  if  /(T)  has  the 
form  2n;cos  (j±\)r,  where  ±  V^T  X  are  the  characteristic  exponents  of  the 
homogeneous  equations,  then  the  particular  solution  has  the  form 


It  is  understood  that  a,  6,  c,  d,  and  j  arc  integers  in  Theorems  IV  and  V. 


132  PERIODIC    ORBITS. 

72.  Integration  of  the  Differential  Equations.  —  It  will  be  convenient 
hereafter  to  use  as  notation  for  the  fundamental  set  of  solutions,  (82)  and 
(83),  of  the  equations  of  variation 

P  =  Aa2(r)+JBa1(r)+Ca3(r)+Z)[a4(r)+Ta3(T)],  1 


where  the  a  and  y-i  unctions  are  characterized  as  follows  : 

UJ(T)  involves  only  terms  of  the  form    cos  [(2n+l)±x]  r, 
Tl(r)         "          "          "         "         "         sin[2n         ±X]r, 

ttl(r)         "          "          "         "         "         sin[(27i+l)±X]r, 
72(r)         "          "          "         "         "         cos[2n         ±X]T, 

O,(T)         "          "          "         "         "         sin[2n         +0]  r, 
74  W         "          "  cos[(2n+l)+0]r, 

a4(r)          "  "  "          "          "          cos[2n         +0]r, 

73  (r)         "          "          "         "         "         sin[(2n+l)+OJT. 

It  will  be  also  convenient  to  write  the  differential  equations  (50)  for 
p  and  a  in  the  form 


where  all  the  0's  are  periodic  with  the  period  2?r;  02  and  04  contain  only 
cosines  of  even  multiples  of  r;  and  03  contains  only  sines  of  odd  multiples 
of  r.  On  the  right  side  of  the  first  equation  the  coefficients  of  terms 
carrying  odd  powers  of  a  contain  only  sines  of  odd  multiples  of  T,  while  all 
the  other  coefficients  contain  only  cosines  of  even  multiples  of  r.  In  the 
second  equation  odd  powers  of  u  have  coefficients  involving  only  cosines 
of  even  multiples  of  T,  while  all  other  coefficients  contain  only  sines  of 
odd  multiples  of  T. 

The  initial  conditions  are 

p(0)  =  a,  p'(0)=0,  cr(0)=0,  <r'(0)  =  6. 

We  will  integrate  equations  (102)  as  power  series  in  a,  5,  and  f,  with  T  entering 
in  the  coefficients.  We  know  that  these  series  are  convergent  for  any 
arbitrarily  chosen  interval  for  r,  0  <*  r  ^  T,  provided  |  a  |  ,  1  5  ,  and  |  e  |  are 
sufficiently  small.  The  equations  of  variation  involve  the  period  27T/X. 
The  solutions  are  not  periodic  unless  X  is  rational.  Hence  the  constants 
upon  which  X  depends  must  be  chosen  in  advance,  so  that  X  shall  be  rational. 
We  will  suppose  then  that  X  =J/K,  where  j  and  K  are  relatively  prime  integers. 
Then  the  first  two  solutions  of  the  equations  of  variation  are  periodic 
with  the  period  2  KIT. 


ORBITS   ABOUT   AN   OBLATE   SPHEROID.  133 

Since  p  and  a  are  expansible  in  po\ver<  of  a.  ft.  ami  f,  \vc  may  write 


The  differential  equations  for  the  ptlt  and  <TO«  are  obtained  by  substituting 
these  expressions  in  (102)  and  equating  the  coefficients  of  similar  powers  of 
the  parameters. 

Coefficients  of  a.     The  coefficients  of  a  are  defined  by 

pr«,+*.Pi«+»i*»  =  0,  ff"loo+6t  <rlM+OiPw  =  0.  (103) 

The  solution  of  these  equations,  which  are  the  same  as  the  equations  of 
variation,  is 


«(T)  +D[ytM 
In  order  to  satisfy  the  initial  conditions  we  must  have,  at  T  =  0, 

Pioo=1>  Pioo  =  0>  0"ioo  =  0»  *ioo  =  0 

From  these  conditions  we  find  that 


The  solution  of  these  conditional  equations  is 

(105) 


A 

A  = 
Hence  the  solution  of  equations  (103)  takes  the  form 


Coefficients  of  5.    The  terms  of  the  first  degree  in  6  must  satisfy 

Pb,.+«iA«.+«.»«o  =  0,  ffiu+0tffm+6tpm  =  0.  (107) 

These  equations  are  the  same  as  (103),  and  from  the  initial  conditions  we 
must  have,  at  r  =  0, 


The  solutions  of  equations  (107)  are  therefore 

r)  +  ra1(r)], 
r)  +T  y<(r)}, 


where 

Am         «4(0)  A2. 

^010=      -  ^010 


134  PERIODIC    ORBITS. 

Coefficients  of  e.     The  differential  equations  for  these  terms  are 

Pan  +  02  Pooi  +#3  ffm  =  0<m  >  crooi  +  ^4  ^ooi+^s  Pooi  =  0- 


The  right  member,  6m,  is  a  periodic  function  of  r  with  the  period  2ir. 
Furthermore,  it  involves  only  cosines  of  even  multiples  of  T.  Consequently, 
by  Theorem  II  of  §71,  the  solution  has  the  form 

r), 
r),| 

where  a  is  a  constant  depending  on  dm  ;  a5  (T)  is  a  cosine  series  involving 
only  even  multiples  of  r;  and  75(r)  involves  only  sines  of  odd  multiples  of  T. 
From  the  initial  conditions  it  follows  that  pm  ,  <rwl  ,  p'm  ,  and  a'mi  all  vanish 
at  r=0.  On  determining  the  constants  of  integration  so  as  to  satisfy  these 
conditions,  the  solution  is 


c, 

[73(r)+T74(T)]+76(r),  j 

where 


A 
a6  (T)  =  a6  (T)  -  aa4  (T)  ,  76  (T)  =  y,  (r)  -  ay3  (T)  . 

It  will  be  seen  at  the  end  of  §73  that  the  value  of  a6(r)  for  T  =  0  plays 
an  important  role,  and  it  is  necessary  for  us  to  verify  that  it  does  not 
vanish.  By  hypothesis,  a5(r)  is  the  periodic  part  of  the  solution  for  p^,  in 
the  differential  equations  (109).  Let  us  put  in  these  equations 

Pooi  =  <P  (T)  +ara3  (T),  trm  =  \f/  (T)  +«T74  (T), 

where  <p  and  ^  are  the  periodic  parts.     We  find 


or,  using  the  explicit  values  of  a3(r)  and  74(7-), 


In  these  last  equations  we  have  put 


ORBITS  ABOUT  AN   OBLATE   SPHEROID.  135 

Let  us  put  now 


mid  integrate  as  a  power  series  in  n,  having  in  mind  that  <p  and  ^  must  be 
periodic.     We  find 


+*i  =  (3+2o,)  /38i 


Since  ^,  is  periodic  we  must  put  aa=  —3/2,  and  then,  after  integrating,  we 
have  ^,  =  c,  sin  r—  1/2  ^c0sin2r.     From  the  coefficient  of  MI  we  obtain 


terms. 
Since  ^,  is  periodic  we  must  take  c,,  equal  to  zero.    Therefore 

V  =  Of  00  =  1  +  power  series  in  /«*,  o  —  -   -  +  power  series  in  /**. 

Consequently 

O,(T)  =  O,(T)—  aa«(T)  =  l+power  series  in  n*,  (112) 

which  does  not  vanish  for  T  =  0. 

Coefficients  of  a1.     The  terms  of  the  second  degree  in  a  are  defined  by 

pk+»«ft«+tooo  =  fl»,  <4>+04P«.+0,P»>  =£«•>,          (H3) 

where  the  right  members  have  the  following  expressions: 


By  the  initial  conditions  p^,  am,  and  their  first  derivatives  vanish  at  r  =  0. 
Since  the  equations  (113)  are  linear,  the  solutions  have  the  form 


=  A  O,(T)  +B  o,(r)  +  CO,(T)  -f-Z>  [o4(r)  +TO,(T)] 


(114) 


136  PERIODIC    ORBITS. 

Upon  imposing  the  initial  conditions,  we  find 


.  _  4-43          .,„,   ,  4 

«  A  ~    ^100      I 


A  ~          100      I  A  "    -^  100  -^  100 

,  ^(0)a4(0)-fr(0)[74(0)+7i(0)]  A  001 

••Ml  > 


L100 


, 

I 


A 


L.  -.  ,  «, 

A  -"-100   ) 

where 

A-at(0)[74(0)+7l(0)I-7i<0)«»(0). 

On  substituting  these  values  in  (141),  we  have  for  the  solutions 

P^A^x^+A^A^x^+A^x^r),   } 
o-»  =  ^lS  y.(r}+A^A?Mr}+A™  y,(r),    J 
where 


Tl(r) 


and  similar  expressions  for  X2,y2,x3,  and  i/3  ,  the  values  of  which  we  shall  find 
we  do  not  need.  The  properties  of  ^  and  y^  are  known  with  the  exception  of 
(p,  and  \f/i,  which  we  will  now  investigate.  The  functions  ^  and  ^  are 
those  portions  of  the  solution  of  the  differential  equations  which  depend 
upon  the  coefficients  of  A^-  These  coefficients  are  homogeneous  of  the 
second  degree  in  a2(r)  and  T^T). 

In  Rm  and  S2W  the  expressions  6200,  0no,  and  6^  contain  only  cosines  of 
even  multiples  of  r;  &M,  6m,  and  ~o~m  contain  only  sines  of  odd  multiples 

00 

of  T;  a2(r)  has  the  form  a2  =   2  an  cos[(2n+l)±X]  T;  y^r)  has  the  form 

n=0 

00 

7i  =   2  bn  sin  [2w±X]  T.      Consequently,  so  far  as  the  coefficient  of  ^[o02  is 

n=0 

concerned,  R200  and  S20B  have  the  form 

Rm=  S  a™  cos2wr+  2  a®  cos  [2n±2X]  T, 

n  =  0  n  =  0 

>S200=  2  61"  sin(2w+l)r+  2  6®  sin  [(2n+l)±2X]  T. 

n=0  n=0 


ORBITS   ABOUT   AN    OBLATE    Sl'HKHtUK  137 

By  §:*<)  terms  involving  multiple*  of  XT  give  rise  only  to  periodic  term- 
in  the  solution.  By  §31  those  part*  of  /,'..  and  >',,.,  which  are  independent 
of  X  give  rise  to  irnn-  in  the  solution  which  have  the  form 

P  =  Pi(r)+r'\T*i(T),  <r  =  Pi 


where  />,(T)  and  p,(r)  are  periodic  with  the  period  2*.      Consequently  the 
function-  J-,(T)  and  J/,(T)  have  the  form 


/'    r)+Clra,(r),  V.M  =P,(T)+c,ry4(r),          (116) 

where  I\  (V)  and  P,(T)  are  periodic  with  the  period  2*7r. 

of  aS.    The  differential  equations  for  these  terms  are 

»~5U»  (H7) 


where 

7?] 


and  SIIO  is  obtained  from  /?110  by  replacing  0,,»  by  (?(J»  . 

The  functions  /?,,„  and  Sm  differ  from  ^  and  S,M  only  in  the  con- 
stants Atlk  .  The  initial  conditions  impose  the  same  conditional  equations. 
Consequently  the  solutions  differ  only  in  the  constants  Attt  ,  so  that  we  can 
express  them  at  once  without  computation  in  the  form 

^0xl(T\  1 
j/1(r),J 

where  the  xt(r)  and  ?/,(T)  are  the  same  functions  of  T  as  in  (115). 

Coefficients  of  S1.    By  symmetry  with  the  coefficient  of  a1,  it  is  seen  that 

A.  =  ^C'  *,  W  +A£  A%  x,  (r) 


(119) 
<*<*>  =  A™  yl  (T)  +A™0  A™0  y,  (r)  +  A™  y>  (T). 


Coefficients  of  £.  Since  the  coefficients  of  the  first  powers  of  a  and  6  are 
homogeneous  of  the  first  degree  in  the  A's,  the  coefficients  of  a*,  06,  and  61 
are  homogeneous  of  the  second  degree  in  the  A's.  The  coefficients  of  the 
first  power  of  «  are  not  homogeneous  in  the  A's;  hence  the  coefficients  of 
the  second  power  are  not  homogeneous.  But  if  the  functions  a,  and  7, 
were  zero  the  coefficient  of  the  first  power  of  «  would  be  homogeneous,  and 
therefore  the  second  also.  By  symmetry,  therefore,  we  can  at  once  write 
down  the  terms  involving  the  A 's  to  the  second  degree.  To  these  must  be 
added  terms  in  the  first  degree  in  the  A's,  and  one  term  independent  of 
the  A's. 


138  PERIODIC    ORBITS. 

The  differential  equations  for  these  terms  are 

//  __  f- 

Po02    I    ^2P()02 ~T~ "3*^002  =  -*^002  >  °0 

where 

Rm  =  02ooPooi  +  flnoPooi^ooi  +  0o20°oo 
2      i  ~a  I  ~a       2 


>  =  Sm,  (120) 


The  terms  involved  in  Rm  are  shown  in  the  following  table,  where  the 
coefficients  of  the  constants  given  in  the  first  line  are  the  products  of  the 
functions  in  their  respective  columns  and  the  functions  of  the  same  line 
in  the  last  column.  Thus,  one  of  the  coefficients  of  A™  is  a2  yj  6m ,  and 
this  coefficient  comes  from  the  expansion  of  p^  <rm  . 


Origin  of 
term 

-(1)2 
A001 

001          001 

4(2)2 

4S 

< 

1 

Multi- 
plied by 

4 

Pool 

ai 

2o,(Ta,+o() 

(ra,+a,)» 

2o,o, 

2a,(ra,+a.) 

aj 

02M 

POOI^MI 

a,  7, 

O,(T  T4+7i) 

(r  «,  +  «,)  (T7.+7.) 

7,0,  +O.TI  •  ^  (r*%?«J 

a,  7, 

"no 

<r'M, 

7? 

27i  (T7.+7,) 

(r  7<+7,)' 

27,7,       ;     2(TT.+7.) 

72 

*OM 

P»i 

>, 

(T0,+o4)             a,              S101 

For  the  Sm  it  is  necessary  in  the  above  table  only  to  change  the  6i}t 
into  Bt]t  in  the  last  column. 

The  solutions  of  equations  (120)  can  be  expressed  in  the  form 


(r 


(121) 


where  x^  xt,  x3,yl}  y2,  and  ?/3  are  the  same  functions  as  in  (115). 

The  coefficients  of  A^  in  the  differential  equations  (120)  are  homo- 
geneous of  the  first  degree  in  a^  and  7j  ,  every  term  of  which  involves  the 
first  multiple  of  XT.  Hence  the  solutions  for  these  terms,  by  Theorem  IV, 
§71,  involve  non-periodic  terms,  and  we  can  write 


)  +C2Ta1(r)+c3  TO,(T), 


(122) 


where  P3  and  P4  are  periodic  with  the  period  2  KIT. 

It  is  seen  from  the  table  that  x^(r)  and  y6(r)  do  not  involve  the  X. 
They  have,  therefore,  the  form 

x6(T)=P6(T)+c4Ta3(T),  y^T^PM+CiTy^T).  (123) 

It  will  be  verified  at  the  bottom  of  page  141  that  we  do  not  need  to 
know  the  character  of  x6(r)  and 


(iKWTS    ABOUT   AN   OBLATE    Sl'HKKOID.  139 

Cocffici<  N/.S  i  if  at.     Those  tcnn-  >:itisfy  tlic  <liflVrenti:il  (-(illations 

Pioi  +  #1  Pioi  +  0i  ff  101  =  #101  ,  ff'n  +  94  »  IM  +  0|  Pin  ~Sm  ,          (124) 


where 


=  OHM  [2  p 


0  [p1 


r,  J  +  8m  [2  <rloc  <rm]  . 


I'll,    following  table  for  Rim,  constructed  like  that  on  page  138,  shows  the 
character  of  the  terms  entering  into  these  expressions: 


<  >ri(!in  of 
term 


,|(1)    ,.(1) 
100^001 


.(i)  j<»  , 

•"IOO  ^001  T 

"    lfM\     *        IHt  1 

*»W»          WII 


<Z)        A  (») 

100  ^ooi 


I") 
100 


9 


Mulli- 
pUedby 


PmPm 
'm'm 


In  order  to  obtain  Sm  it  is  necessary  only  to  change  the  0f/»  into  0,lk  in 
the  last  column  of  the  table. 

This  table  shows  that  Rm  and  Sm  differ  from  Rm  and  Sm  only  in  the 
constants  A,lt .  Since  the  initial  conditions  impose  the  same  conditional 
equations  as  for  the  coefficient  of  «*,  the  solution  has  the  form 


A™  Vl  (r)  -f  [A 


(r) 


(125) 


where  the  X,(T)  and  y,(r)  are  the  same  functions  as  in  (121). 

Coefficients  of  6«.    These  coefficients  can  be  obtained  by  symmetry  from 
the  coefficient  of  a«,  and  are 


i  xi(r 


A" 


y4  (r 


yt  (r). 


(126) 


This  concludes  the  computation  of  all  terms  up  to  the  second  order 
inclusive  in  a,  5,  and  e.    It  is  not  necessary  to  carry  the  computation  further. 


140  PERIODIC    ORBITS. 

73.  Existence  of  Periodic  Orbits  having  the  Period  2 KIT. — We  have 
chosen  the  initial  conditions  so  that  at  r=0  the  particle  is  crossing  the  p-axis 
orthogonally.  It  is  obvious  geometrically  that  if  at  any  future  time  it 
again  crosses  the  p-axis  perpendicularly,  the  orbit  will  be  a  closed  one  and 
the  motion  in  it  will  be  periodic.  The  conditions  that  the  particle  shall 
cross  the  p-axis  perpendicularly  at  r  =  T  are  that  at  this  epoch  p'  =  a-  =  0. 

The  equations  of  variation  have  the  period  2  KIT.  Therefore  we  shall 
choose  T  =  KIT.  Since  p  is  an  even  series  in  r,  and  a  is  an  odd  series,  all  the 
purely  periodic  terms  in  p  and  a  are  sines,  and  consequently  vanish  at 
T  =  KTT.  The  terms  which  do  not  vanish  must  carry  T  as  a  factor.  The 
conditions  for  periodicity  give  us  the  two  equations 


p'  (KIT)  =  0  =  am  a + a010  5 + am  t.  -f  a2oo  <*2 + ano  a  5 + am  52 + aon  5  e 
-l-a^ae+aooje2^-   •  •  •  , 

<r  (KTT)  =  0  =  bm  a+bowd+bmf+bm  a2+6110  a5+602052+6011 5e 

ae+6002e2  + 


(127) 


where  ai)k  and  bilk  are  the  coefficients  which  were  computed  in  §72.     Their 
explicit  values  are  as  follows  : 

n       -  A  (2)  11  ft      —  A  (2>  11  n     —  A  (2>  11 

uioo  —  -'MOO  w>  uoio      -^oic  «*i  wnoi  —  -^ooi  u> 

J,        _    A  (2)    ,.  h        —    A  <2>    ,,  },        —    A  (2)     „ 

"100  ""-^MOO   Vi  "010         "<M  PJ  "001  ~-^001    V! 


OO 

(2) 
100  ""-MOO      i  "010         "<M      J  "001 

(1)2  ~T__    ,<  (1)     J  (2)    ~T__  J  (2)2 


_    ,4  ( 

•a- 


r          _   J(l)2          I/Id)     yj(2)          4-Xle)2ii7 

"200  "-^loo  t/i  i  -"-100  -^100  at  \  -"-im  Us  > 

A  (2)  -I-  A  (2>  A(in  ^r  4-9  A™ 
-'1oio~-':lioo  -^oioJ  *t  '  ••Am 


^     _oj(»    /Id)  ^"j.fJt1)  /d(2)_L/l(2)    ,4  (DlTT  in  /l  (2)    /«(2)  17 
"no  —^^100  -^010  i/iT^L^ioo  -^OIOT-AIOO  -^oioJ  4/2  i  ^-^loo  -^010  i/3  > 

„     _  A  (1)2  r~  i  j  (i)  x  (2)  ~  I  A  (2)2  ~r 

Wo2o~-'loio  •*/i~-^1oio  -^oio  •''si  ^010  -^s  ? 
h       -A  (1)2  i7-4-  A  (1>  xl  (2)  i74-  A  C2)277 

"020  —-^010  £/l    1^-^010^010  i/2    I    -^010  i/3    ) 

—  9  Am  Am  r~-l-r/l(1)  /1(2)  4-4(2)  /l(I)lr"4-9  4  (2)  A  (2)  r"-l-  /4(1) 
"loi^^-^-ioo-^ooi^i  i  L-^1  loo-^ooi  i  -^loo-^ooiJ-*-!  i  ^-^loo-^ooi^a  i  -^IOO 

_o/i(i)  /(i)7    i  r  /id)  j<2>    id  (2)  /id)  177-4-9  /4(2)  xl<2>  i74-  /4U)  TT-)-/!'2'  ?7 

~^O/1OOlJi/2    I    ^^lOO-^OOl  i/3  ^^lOO^T^-^1  100i/5 


,,    _o  /jo)  /4'1)_i_r/4(»  /i<2)  _i_/4(2)  xdMio  /i(2)  4(2)     _i_  j 

"Oil  —  ^-^OlO-^OOl^l    I    L-^OlO-^MlT-^olO-^OOlJ^    I    •"-'1010-'1001't'3    I    -'1 


__  _ 

010-'1001't'3    I    -'1010't'4T^-'1010-l'5  ! 


_    A  (D2  ~_l      ,4  (1)      A  (2)   ~     I      /(  (2)2  7T_I_   yj  (»  ~     I      A  (2) 

"002     -"^ooi  *i  t  **ew  -^ooi  *a  i  -"ooi  *»>  •"•001*4  <***w 

0(X)2  =-^001  2/1    I    -^001  -^001  2/2~)~-4o01  2/3~t~-4o01?/4~f~-4oOl 

where 

U  =  KTT~  ,  V  =  Kirjt,  Xt  =  ^  ,  yi  =  y{  atT  =  K7T.  (129) 

UT  '  ar 


OKBIIS    AUDI  T    AN    OHLATK    H'HKKOII). 


141 


Let  us  solve  the  first  equation  of  1 127)  for  «  as  a  power  series  in  a  and  6. 
\\ V  obtain 

e  =  e10a+«Bl«+«wat+«,,a«+<-*l+   ...  §  (130) 

where  the  coefficients  t,,  have  the  values 

_  a,o»  oJo,- a100  a,, 
ai>i 

(131) 


f   -- 

"10  —  >  «*>  ' 


"•: 

lo,aM,  a,,, 


<, 
Solving  the  second  equation  of  (127)  for  e  in  terms  of  a  and  6,  we  obtain 

«  =  «iia+«^6+^a1+e^a5+«^a,+   •  •  •  ,  (132) 

where  the  e^  have  the  same  expressions  in  the  6(>t  as  the  ««/  have  in  the  a,Jt  . 
Upon  subtracting  (130)  from  (132),  we  have 

0  =  S-«1o]a+[^-eoi]5  +  S;-ei(,]a!-r-S;-€1Ila6  +  [<--e(,]51+-  •  •  (133) 
We  must  examine  the  coefficients  of  this  series.     The  first  two  are 


QOIO     Oaio 

^         i.      =  jm 

Oooi     Oooi     ^001 


QIOO  _  OIOQ     •"•  IPO  V -A  IPO  P. 

Oooi     Oooi     -"001 W     /looi  P 

Both  of  the  linear  terms  therefore  vanish. 

The  computation  of  the  second  degree  terms  is  somewhat  more  compli- 
cated. It  will  simplify  matters  somewhat  if  we  observe  that  the  e^  are 
the  same  expressions  in  v  and  j/j  as  the  «,»  are  in  u  and  x< .  It  will  therefore 
be  sufficient  to  compute  one  and  derive  the  other  from  it.  On  substituting 
in  the  expression  for  e^  in  (131)  the  values  of  the  aljt  from  (128),  we  get 
,1 


-..  r  A  wt  <la)*;ri-l_>l 
"3^1        °°          '     " 

— 

|- 

•**ooi  L 


<l> 


A  a)*  ??  -4-  A 
' 


tt)t 


' 


100  -100  - 


100  '^001  - 


<*>  ?4 


Xl  -U 
~    r 


A  o)»  A  <»     « 

T  ^100   -^00 


4  _       4 

-^ 


4- 

1  —    r 


(1)1    J  (») 


tt)t 


a>* 


?*  _i.  A tt>1  ^i\ • 
•u         —     -'M         loouJ 

On  forming  the  sum  of  these  three  expressions,  there  results 


(135) 


(fil 


100 


100001 


u 


the  coefficients  of  o^/u,  x,/u,  and  xju  being  identically  zero. 


142  PERIODIC    ORBITS. 

On  changing  the  ^  into  ^  and  u  into  v,  we  get  —  &T0  .     Hence 

1C  2       —  "  — 

|r/|CU      AW  _   AW     J">1     fS_al 
€20        C20  •     4  <2>3  1      ^100  ^OOl         ^MOO^OOl 

•^001  L  <*        " 

~       |_      ,1( 

1 


^6   _  |/6~|  1 

LU         v  \  \ 


But  f-1  -  ^1  and  f-6  -     1  vanish  since 
LM       yJ          LM       yJ 


as  is  readily  seen  from  (116)  and  (123).     On  referring  to  (122),  it  is  also  seen 
that  |^4  —  fel  does  not  vanish,  but  is  equal  to  c,|-  —  —  -^1  Hence 

LU        VJ  LU  dr         VJrmxr 


[—         "1 
_  ,      _  _  - 
20      20J 


™  \Am  A(2)  —  A™ 
100  L-rt-  100  ^001     -^100  - 

~ 


ln7       H7~\ 
J    Xj  _  #4 

lu~v\ 


(136) 


Without  repeating  the  details  of  the  computation,  we  find  similarly 

r/ia)  4<2)     4(2)  4(1Mr^    ,, 

L-^L010-a001~'ft010-^1OOlJ      *4_i/ 

Lw    e 


t-l 
,       _  _ 
608    e<BJ" 


J  (2)    r  A  (1)      /I  (2)   _    J  (2)      A  (1) 

_  _  -^010  L-^oio  -^001  _  -fl-oio  -^ooi 

" 


r-    -'  -, 

\  Xt  _  i/4 

Li     PJ 


(137) 


On  substituting  in  (133)  the  values  obtained  for  the  coefficients,  we  find 
that  the  second  degree  terms  in  a  and  5  are  factorable,  giving 


(138) 


-"-      ~ - 


There  are  therefore  two  solutions  for  6  as  power  series  in  a. 

On  substituting  the  two  solutions  of  (138)  for  6  in  (130),  we  find  the 
two  corresponding  values  of  e.     We  thus  obtain  the  two  solutions 


A  (1)     A  (2)  A  (2)     A  (1) 

71 100  ^001         •"  100  -"-Ml 


f  =  — 


8=- 


A  <1)     J  (2)  _  4  (2)     4  CD    •*    1 

-^oio  -^ooi      -^oio  -^ooi 

7s  \W      i 

a5(0)-aa4(0)a  ' 

X  (1)     A  W          A  W     A  CD 

•"-  100  -^-010      -^100  -^010      1 

•  —      /n\  a               '    , 
a6(0) 

1 

4(1)  4<2>       j  (»  4<i>  a  1 

•^010  **«B       •"•KM  •*981 

„    1 

a5(0)-aa4(0)a 
^S 

J  (2)                  J  (2) 

-Aooi          -"-001 

Oe(O)  °                                    ] 

=  ^-a+..., 
=       0-a+   •   •   •   , 

(139) 


(140) 


ORBITS   ABOUT  AN   OBLATE   SPHEROID.  143 

where  a,  and  y,  are  the  quantities  defined  in  (111),  and  -yi(0)  is  the  value 
of  dyt/dr  for  r  =  0.  It  was  shown  in  (112)  thata,(0)  is  distinct  from  zero, 
and  in  (79)  that  o,(0)  i<  equal  to  unity.  Thus  one  solution  for  t  begins 
with  the  first  power  of  a,  while  the  other  certainly  does  not  begin  before 
the  second,  but  in  both  solutions  5  begins  with  the  first  power  of  a. 

74.  Construction  of  the  Solutions  with  the  Period  2 KIT. — We  have 
proved  the  existence  of  series  for  p,  a,  and  t  proceeding  in  powers  of  the 
initial  value  of  p,  which  we  will  now  denote  by  e*.  The  series  for  p  and  a  are 
periodic  in  r  with  the  period  2KW,  and  since  this  condition  holds  for  all  values  of 
e  sufficiently  small,  each  coefficient  separately  is  periodic.  The  series  for  p 
is  even  in  T,  and  the  series  for  a  is  odd  in  T.  These  series  have  the  form 

p  =  p,e+p,e1+p,e'+  •  •  •  , 


(141) 

i  •    i  _«    i 

e  = 

We  shall  substitute  these  series  in  equations  (102)  and  integrate  the  coeffi- 
cients of  the  powers  of  e  in  order,  and  determine  the  constants  in  such  a 
way  that  p  and  a  shall  be  periodic  and  shall  satisfy  the  initial  conditions 

p(0)=e,          <r(0)  =  0,  p'(0)=0,         *'(G)  =  v, 

where  v  is  a  constant  which  will  be  determined  in  the  process. 

On  substituting  the  series  (141)  in  the  differential  equations  (102),  we 
find  for  the  coefficients  of  the  first  power  of  e 

W  9    \    t\  \    t\  f\  / i  A  (\\ 

By  the  condition  of  orthogonality  p  must  be  even  in  T  and  a  odd  in  T,  and 
the  solution  complying  with  these  conditions  is 

._  _  /— \  _i_  _  /— \i 

(143) 

where  OJ(T)  contains  only  cosines  of  even  multiples  of  T,  and  yk(r)  contains 
only  sines  of  odd  multiples  of  T.  In  order  that  this  solution  shall  be 
periodic  it  is  necessary  and  sufficient  that 

Upon  substituting  this  value  of  D(U,  the  solution  (143)  becomes 

Pi  =  Aw  a,[(T)+«1a,(T)—  aa4(r)]  =  A     O,(T)+«,  O,(T), 

[          (144) 
ff1  =  Aw  7i  [(T) +€,7i  (T)  =  a  7,  (T)  ]  =  A    -y,  (T) +«i  T«(T)- 

It  remains  to  impose  the  initial  condition  that  p,  =  1  at  r  =  0,  which  gives 

l-^ii+c.a.,  (145) 

where  a,  and  a,  denote  the  values  of  a,  and  a.  for  T  =  0. 

The  reason  for  changing  from  a  to  «  is  that  this  parameter  corresponds  to  the  eccentricity  in  the 
two-body  problem. 


144  PERIODIC    ORBITS. 

Coefficients  of  e2  .     The  coefficients  of  e2  satisfy  the  equations 

2  P2+03  °2  =  #001  «2  +  0101  «1  Pl  +  0200  Pl+0110  Pi  ^1  " 

io  ZT      2  i  2T 

0-2  +  03  p2  =  0200  Pi+t/no  Pi 

Every  term  of  R2  and  &  contains  either  Aw,  (1  ,  or  e2  as  a  factor.     Arranged 
in  this  manner,  we  have 

[  Bm  a*+0110  a,  Tl+0o2o  7i  ] 
e1  [20200  a2  a6+^110(71  a^a-j  76)+  20020  7, 
76       020  76 


7i 

+  «?  [  0200  ^6  +  ^110  «6  76  +  ^020  7o  ]  • 

In  order  to  understand  the  character  of  the  solution  of  equations  (146),  we 
must  examine  the  character  of  722  and  S2  .  The  coefficient  of  A(m  in  both  722 
and  $2  is  homogeneous  of  the  second  degree  in  a2  and  yt  .  Its  expansion  there- 
fore involves  terms  carrying  2Xr  and  terms  independent  of  XT.  By  §30,  the 
solution  for  the  terms  in  2  XT  is  periodic.  The  terms  independent  of  XT  are 
cosines  of  even  multiples  of  r  in  R2  ,  and  sines  of  odd  multiples  of  T  in  S2  . 
These  terms  have  the  same  character  as  those  in  the  coefficients  of  e,  and  e2  , 
and  will  be  considered  under  the  discussion  of  those  terms. 

The  coefficients  of  Amet  in  both  R2  and  S2  are  homogeneous  of  the 
first  degree  in  o2  and  7t  ,  all  terms  of  which  carry  the  first  multiple  of  XT. 
By  Theorem  IV,  §71,  the  expression  for  p2  will  carry  the  term  TO^T),  and 
for  <r2  ,  the  term  T72  (T)  .  Non-periodic  terms  of  this  character  do  not  arise 
elsewhere  in  the  solution.  Hence,  in  order  to  avoid  them,  we  must  take 
either  ,A(1)  =  0  or  ^  =  0.  If  we  choose  ^la)  =  0,  then,  by  (145),  ex  is  determined 
and  has  the  value  e1  =  l/o^,  thus  agreeing  with  the  first  solution  (139)  of 
the  existence  proof.  But  if  we  choose  ^  =  0,  so  that  by  (145)  Am  =  1/c^, 
we  are  in  agreement  with  the  second  solution  (140)  of  the  existence  proof. 
We  will  commence  by  developing  the  first  solution,  in  which 


FIRST   SOLUTION. 

Since  ^4a)  =  0,  all  terms  in  .R2  and  $2  which  carry  XT,  or  any  multiple  of 
XT,  vanish.     There  remain 


a6  76+0020  76+0wi  a6]  +  6,  0,01  , 


OHH!I>     Mini    I      \N    OBL.VIK    Sl'MKUiUD.  145 

We  have  &]><> 

OL*  CLt  O« 

It  follows  from  (147)  that  R.  contains  only  cosines  of  even  multiples  of 
T,  and  <S,  contains  only  sines  of  odd  multiples  of  T.  Since  p,  is  an  even 
function  of  T  and  a,  is  ;m  odd  function  of  T,  the  solution  is 

i.a*«.fetf.l 

(149) 

In  this  solution  the  terms  are  grouped  according  to  their  origin.  The 
first  two  terms  are  the  complementary  function.  The  third  arises  from 
the  terms  carrying  e,  as  a  factor.  The  fourth  arises  from  the  terms  having  e, 
as  a  factor,  a,  is  a  constant  depending  upon  the  coefficients  of  «J  in  the 
differential  equations,  and  O,(T)  and  yt(r)  are  the  same  functions  as  in 
the  coeflieient  of  the  first  power  of  e,  TJ,(T)  and  $*,(T)  are  periodic  functions 
of  T  with  the  period  2w,  and  so  constituted  that  ry,(r)  contains  only  cosines 
of  even  multiples  of  T,  and  f,(r)  contains  only  sines  of  odd  multiples  of  T. 
1 1 1  order  that  p,  and  a,  shall  be  periodic  we  must  have 

which  makes 

A.  =  A  ™  a,  (T) + e,  o,  (T)  +  n,  (T)  -  O,  at  (T)  , 

(150) 


In  order  that  we  may  satisfy  the  initial  conditions,  we  must  have  p,  =  0  at 
T  =  0,  which  determines  «,  by  the  equation 


a, 


I  '  is  obvious  that  At  ,  which  so  far  is  arbitrary,  must  be  zero,  for  in  the 
coefficient  of  e*  it  will  give  rise  to  terms  involving  the  first  multiple  of  XT. 
All  such  terms  will  carry  Am  as  a  factor;  hence  to  avoid  non-periodic 
terms  of  this  character,  we  choose  A(1>  =  0.  Anticipating  this  step,  we  have 


(151) 


so  that  Pt  contains  only  cosines  of  even  multiples  of  T,  and  <rt  contains  only 
sines  of  odd  multiples  of  T. 

It  only  remains  to  show  that  this  process  of  integration  can  be  carried 
on  indefinitely.  On  assuming  that  up  to  and  including  p,_,  and  er,_,  every 
p,  and  a,  is  periodic  with  the  period  2*,  and  that  the  p,  contain  only  cosines 
of  even  multiples  of  T  and  the  <r,  only  sines  of  odd  multiples  of  T,  except  that 


146  PERIODIC    ORBITS. 

p(_!  contains  the  term  ^"""o-jW  and  o^  contains  the  term  A^^y^r),  it 
will  be  shown  that  the  same  conditions  will  obtain  for  the  next  succeeding 
step.  For  p(  and  a-f  we  have,  from  the  differential  equations  (102), 


pf 


+26 


m 


(152) 


From  the  properties  of  the  differential  equations  it  is  readily  seen  that 
<£(  contains  only  known  terms  all  of  which  are  cosines  of  even  multiples 
of  T,  and  that  SF,  contains  only  known  terms  all  of  which  are  sines  of  odd 
multiples  of  T.  The  coefficients  of  Au~  "  are  homogeneous  of  the  first  degree 
in  o-j  and  yl  ,  and  consequently  each  term  involves  a  first  multiple  of  AT. 
They  give  rise  to  non-periodic  terms  of  the  form  TO/T)  and  TJ^T)  in  the 
solution.  They  carry  A"'  "  as  a  factor,  and  since  terms  of  this  type  arise 
nowhere  else,  we  can  make  them  disappear  only  by  putting  A"~'"  =  0.  The 
solution  for  (152)  then  has  the  form 


a3r,      . 

l  (loo) 
)  +T3(r)]  +  [f4(r)  +airy,M]+ei[y6(r')  +arT4(r)],  J 


where  ^(T)  and  f((T)  are  periodic  with  the  period  27r,  and  where  by 
Theorem  II,  §71,  ^(r)  contains  only  cosines  of  even  multiples  of  T,  and  £{(T) 
contains  only  sines  of  odd  multiples  of  r. 

In  order  that  pt  and  a-t  shall  be  periodic  it  is  necessary  and  sufficient 
that 

D("=-at-d€t, 
which  makes 

=     (i)-,  1 

.  j 

From  the  initial  conditions  we  must  have  pt  =  0  at  T  =  0,  which  determines 
c,  by  the  equation 


Thus  the  constants  are  uniquely  determined.  The  p1  and  <rt  have  the 
properties  assumed  for  those  having  smaller  subscripts,  and  the  process  of 
integration  can  be  continued  indefinitely.  Every  Aw  is  zero.  Since  no 
terms  involving  the  XT  enter,  the  solution  has  the  period  2ir,  and  the 
orbits  represented  belong  to  the  class  of  generating  orbits  with  which  we 
started.  In  other  words,  we  set  out  with  a  generating  orbit  for  which  the 
initial  distance  was,  let  us  say,  r0  ,  and  we  have  found  another  generating 
orbit  for  which  the  initial  distance  is  rc+e  (e  arbitrary).  There  is  nothing 
surprising  in  this,  for  r0  is  a  function  of  an  arbitrary  constant  /3. 


OKBITS     U«>l    I     \\    OBLATK    SI'IIKK.  UD.  147 

Let  us  suppose  we  h:i»l  started  with  a  definite  value  of  0,  for  example 
00,  which  gives  a  definite  lienerating  orbit  with  a  definite  initial  distance  r0. 
Let  us  seek  now  the  general  ing  orbit  for  which  the  initial  distance  is  r0+e. 
It  .  is  sufficiently  small  we  can  evidently  give  an  increment  «  to  00,  which 
will  increase  r0  by  the  amount  c.  We  have 

'„=/(&),  r.+e-/(A+«). 

Expanding  the  right  member  of  the  second  equation  in  powers  of  e,  we  have 

df 

e" 


which  gives,  by  inversion,  a  series  of  the  form 


Then,  by  substituting  /3  =  /30+cle+cte*+  •  •  •  in  the  generating  orbit  and 
arranging  the  solutions  as  a  power  series  in  e,  we  obtain  the  orbit  in  which 
the  initial  distance  is  r0+e.  As  these  are  the  same  conditions  that  were 
imposed  when  we  sought  new  orbits  through  the  equations  of  variation,  it 
was  to  have  been  expected  that  one  of  the  class  of  generating  orbits  would 
satisfy  them. 

SECOND   SOLUTION. 

We  return  now  to  equation  (146),  and  continue  with  the  second  solu- 
tion, in  which  €,  =  0  and  A(I)  =  l/o,.  From  (82)  it  is  seen  that  a,  =  a,(0)  =  l, 
and  therefore  Am  =  1.  Hence  in  the  second  solution 


.  (155) 

On  using  these  values  of  A(l>  and  «,  ,  J?,  and  St  of  (146)  become 

0,7, 


All  of  the  terms  in  these  expressions  except  e,^,  are  of  the  second 
degree  in  o,  and  7,  .  Therefore  they  involve  only  terms  carrying  2  XT  and 
terms  independent  of  XT,  and  0m  is  independent  of  XT.  In  the  solutions 
the  terms  depending  upon  2  XT  are  periodic.  As  for  the  terms  independent 
of  X,  Rt  contains  only  cosines  of  even  multiples  of  r,  and  St  contains  only 
sines  of  odd  multiples  of  r.  These  terms  give  rise  to  non-periodic  terms  in 
the  solution,  which  has  the  form 


where  ^>,(X,  T)  and  ^,(X,  T)  are  the  periodic  terms  involving  X;  TJ,  and  f, 
are  the  periodic  terms  with  the  period  2ir;  a,  is  the  constant  belonging  to 
the  non-periodic  part;  and  the  coefficients  of  t,  are  the  solutions  depending 


148  PERIODIC   ORBITS. 

on  the  coefficient  of  e2  in  the  differential  equations.     In  order  that  this 
solution  shall  be  periodic,  it  is  necessary  that 

n<2>  _          „  r,. 

U      —  —  Q2  —  fl€2  j 

which  reduces  p2  and  <rt  to 


,  T)+r2(r)-a2T3(r)+62T6(r).         j 

To  satisfy  the  initial  conditions  we  must  have   p2(0)  =0.      Hence, 
since  o2(0)  =  1,  Aw  is  defined  by 

A(2)=-^2(0)-%(0)+a2a4(0)-e2a6(0), 

where  e2  is  a  constant  which  is  determined  by  the  periodicity  condition  on 
the  coefficient  of  e3. 

Coefficients  of  e3.  The  coefficients  of  the  third  degree  terms  are  denned  by 

s  =  S3,  (159) 


where 

Pj 


~  0001  «3  +  01  "  " 


00  PlP2  +  011ok2Pl  +  0-lP2]  +20Q20  (TiO-j  +  0300  P? 
J10  Pi  °"l  +0120  °"l  Pi  +0030  CT1  • 

In  classifying  the  terms  which  belong  to  the  expansion  of  R3  and  S3  , 
we  bear  in  mind  that 

1.  The  &ilt  in  R3  involve  only  cosines  of  even  multiples  of  T,  except  those 

which  are  coefficients  of  odd  powers  of  a  (i.  e.,  where  j  is  odd),  and 
these  involve  only  sines  of  odd  jnultiples.  The  opposite  is  the  case  in 
the  6iik  of  S3  .  If  j  is  even,  the  6iit  involve  only  sines  of  odd  multiples 
of  T.  If  j  is  odd,  the  di]t  involve  only  cosines  of  even  multiples. 

2.  The  terms  independent  of  X  involve  only  cosines  of  even  multiples  of  T 

in  the  expressions  for  p1  and  p2  ,  and  only  sines  of  odd  multiples  of  T  in 
the  expressions  for  <rl  and  cr2  . 

It  is  seen,  then,  that  in  those  terms  of  _R3  which  are  independent  of  X 
only  cosines  of  even  multiples  of  T  enter  ;  and  in  those  terms  of  S3  which 
are  independent  of  X  only  sines  of  odd  multiples  enter.  In  the  process 
of  integration,  therefore,  two  types  of  non-periodic  terms  arise.  First, 
those  coming  from  the  terms  which  involve  the  first  multiple  of  XT,  and 
secondly,  those  coming  from  the  terms  which  are  independent  of  X.  It  is 
important,  therefore,  to  separate  the  various  terms  into  three  classes,  (a)  terms 
independent  of  X,  (6)  terms  involving  first  multiple  of  XT  only,  (c)  terms 
involving  multiples  of  XT  higher  than  the  first. 


ORBITS  ABOUT  AN   OBLATE   SPHEROID.  149 

We  rewrite,  then,  the  differential  equations  (159)  in  the  form 

,  r), 


tlii  ,,,  t,,4K;T, 

where 


The  /,  and  <?,  terras  are  homogeneous  of  the  first  degree  in  a,  and  %  ,  and 
consequently  involve  only  terms  which  carry  the  first  multiple  of  XT;  they 
are  considered  separately  from  other  terms  of  the  same  character,  because 
they  carry  the  undetermined  constant  «,  as  a  factor.  The  solution  for  these 
terms  has  the  form 


p  =  Fl(\,  TH&.TO.CT),  a  =  G1(X,  T)+6,ry,(T), 

where  F  ,  and  G,  are  periodic  and  involve  only  terms  carrying  the  first 
multiple  of  XT;  &j  is  a  constant  depending  upon  /,  and  gl  ,  and  is  distinct 
from  zero. 

The  /t(X,  T)  and  gt(\,  T)  have  the  same  properties  as  /,  and  gl  .  They 
are  considered  separately,  since  they  do  not  carry  «,  in  their  coefficients. 
Their  solutions  may  be  written 


where  F  ,  and  (?,  are  periodic. 

The  /,  (T)  and  gt  (T)  are  independent  of  X,  and  /3  carries  only  cosines  of 
even  multiples  of  r,  while  gt  carries  only  sines  of  odd  multiples  of  T.  The 
solution  for  these  terms  has  the  form 

P  =  FI(T)+6JTOI(T),  <r  =  G,(T)+&,TY4(T), 

where  Ft  and  G,  are  periodic. 

The  /«(/tX,  T)  and  gt(*\,  T)  involve  only  terms  which  carry  multiples 
of  XT  higher  than  the  first.  The  solution  for  these  terms  is  periodic  and 

may  be  written 

p  =  F4(/cX,  T),  er  =  G4OcX,T). 

The  complete  solution  is  therefore 

^[at(T)  +  aTa,(T)]+«1[F1(X, 

,  T), 


,  T). 

All  the  functions  O,(T),  T,(T),  F,(T),  and  G,(T)  are  periodic.  In  order  that 
p,  and  at  shall  be  periodic,  it  is  necessary  and  sufficient  that  the  coefficient 
of  TO,(T)  and  T74(T),  and  the  coefficient  of  TO,(T)  and  ry,(T)  be  zero;  whence 


150  PERIODIC    ORBITS. 

Consequently  the  value  of  e2  is  determined.  In  order  to  satisfy  the  initial 
conditions,  we  must  have  p3=0  at  r  =  0,  which  determines  A(3)  by  the 
equation 


Thus  all  the  constants  are  determined  except  e3  ,  and  the  solution  is 
p3  =  A(3)  a2(r)-fo3a4(r)+e3a6(r)-  |  F,(\,  r)+F2(X,  r)+F3(r)+F4(/cX,  r), 

icX,  T). 


The  constant  e3  will  be  determined  in  satisfying  the  periodicity  con- 
dition for  the  coefficients  of  e\  It  is  obvious  that  this  process  of  integra- 
tion can  be  continued  indefinitely.  The  p3  and  a3  have  the  same  properties 
that  had  been  found  for  p,  and  p2.  It  is  evident  from  the  properties  of 
the  differential  equations  that  these  properties  persist  for  p4  and  o-4,  and 
so  on  indefinitely.  The  coefficient  for  €,_,  ,  in  so  far  as  it  carries  the  first 
multiple  of  XT,  is  always  the  same  as  for  e,  .  Therefore  the  arbitrary  constant 
«,_!  can  always  be  determined  so  as  to  avoid  non-periodic  terms  of  the  type 
TO^T)  and  T72(r).  The  constant  Dc"  of  integration  can  always  be  deter- 
mined so  as  to  destroy  non-periodic  terms  of  the  type  ra3(r)  and  T74(r). 
The  constant  A(n  can  always  be  determined  so  as  to  satisfy  the  initial 
conditions.  The  analysis  of  the  types  of  terms  entering  is  the  same  as  for 
the  subscript  3. 

We  have,  therefore,  a  periodic  solution  with  the  period  2o-  which  does 
not  belong  to  the  class  of  generating  orbits  from  which  we  set  out,  for  the 
particle  makes  many  revolutions  before  its  orbit  re-enters. 

After  substituting  the  value  of  r  in  the  equation 

dv  __  c^ 
dr       r2 

and  integrating,  the  solution  contains  five  arbitrary  (except  for  the  restriction 
that  X  shall  be  rational)  constants  corresponding  to  the  mean  distance,  the 
eccentricity,  the  inclination,  the  longitude  of  the  node,  and  the  epoch.  One 
more,  a  constant  corresponding  to  the  longitude  of  the  perihelion,  is  necessary 
for  a  general  solution  of  the  differential  equations.  The  periodic  orbits 
developed  here  are  special  in  that  they  are  all  symmetrical  with  respect  to 
the  equatorial  plane  of  the  oblate  spheroid. 


CHAPTER  V. 

OSCILLATING  SATELLITES   ABOUT  THE  STRAIGHT- 
LINE  EQUILIBRIUM  POINTS. 

FIRST  METHOD.* 

75.  Statement  of  Problem. — Lagrange  has  shownf  that  if  any  two 
finite  spherical  bodies  revolve  about  their  common  center  of  mass  in  circles, 
then  there  are  three  points  in  the  line  of  these  masses  such  that,  if  infini- 
tesimal bodies  be  placed  at  them  and  projected  so  as  to  be  instantaneously 
fixed  relatively  to  the  revolving  system,  they  will  always  remain  fixed  rela- 
tively to  the  revolving  system.  There  are  also  collinear  solutions  in  which 
only  the  ratios  of  the  mutual  distances  of  the  three  masses  remain  constant, 
but  in  this  chapter  we  shall  consider  only  the  case  in  which  the  distances 
themselves  are  constant.  In  Chapter  VII  the  more  general  case  will  be 
treated.  The  three  positions  which  the  infinitesimal  body  may  occupy  are 
separated  by  the  finite  bodies;  i.  e.,  starting  from  minus  infinity,  the  order 
is  an  equilibrium  point,  a  finite  body,  an  equilibrium  point,  the  second  finite 
body,  and  the  third  equilibrium  point.  It  is  not  necessary  that  one  of  the 
three  masses  shall  be  infinitesimal,  but  we  shall  limit  ourselves  at  present 
to  this  case.  In  Chapter  VIII  it  will  be  shown  that  the  problem  can  be 
generalized  to  n  masses.  There  are  also  solutions  in  which  the  bodies  lie 
at  the  vertices  of  an  equilateral  triangle,  and  the  oscillations  about  these 
points  will  be  treated  by  Dr.  Buck  in  Chapter  IX. 

If  the  sun,  earth,  and  moon  were  so  placed  as  to  satisfy  the  conditions 
for  a  straight-line  solution,  and  if  the  earth  were  between  the  sun  and  moon, 
then,  as  Laplace  first  pointed  out,  the  moon  would  always  be  full,  and  either 
the  sun  or  the  moon  would  always  be  above  the  horizon  of  every  observer. 
But  these  conditions  would  not  be  preserved  unless  the  moon  were  in  a 
position  of  stable  equilibrium.  If  the  position  were  one  of  complete  stability 
and  the  moon  were  slightly  disturbed  from  it,  then  it  would  perpetually 
oscillate  about  the  point  of  equilibrium;  if  the  position  were  one  of  complete 
instability,  a  slight  disturbance  of  the  moon  would  cause  it  to  depart  widely 
from  the  point  of  equilibrium.  In  the  intermediate  case  of  incomplete 
stability  and  also  incomplete  instability,  the  moon  would  either  oscillate 
about  the  point  of  equilibrium,  at  least  for  some  time,  or  it  would  speedily 

•Read  before  the  Am.-rican  M»them*tical  Society.  Juno  28,  1900;  mbrtwct  in  BulUtin  of  tHt  America* 
Mollumatital  Society,  vol.  VII  (1900),  p.  12.  The  Moond  method  is  givan  in  Ch»p.  VI. 

tUgrtmge'B  ColUeUd  Workt,  vol.  VI,  pp.  229-324;  Ti*er»nd'i  Mtcenique  COetU,  vol.  I.  Ch»p.  8; 
Moulton'g  Introduction  to  Celetlial  Mtchanict,  Chap.  7.  Ul 


152  PERIODIC    ORBITS. 

depart  from  it,  according  to  the  character  of  the  disturbance.  If  it  is  given 
such  an  initial  displacement  that  it  revolves  in  the  vicinity  of  the  point  of 
equilibrium  in  an  orbit  closed  relatively  to  the  moving  system,  it  is  called  an 
oscillating  satellite;  for,  as  seen  from  the  earth,  it  oscillates  in  the  neighbor- 
hood of  the  equilibrium  point  in  an  apparently  closed  orbit.  We  shall  consider 
here  the  motion  of  infinitesimal  satellites  oscillating  in  the  vicinity  of  each 
of  the  three  collinear  points  of  equilibrium. 

The  literature  of  oscillating  satellites  is  quite  extensive,  but  in  most 
of  the  papers  the  differential  equations  have  been  limited  to  their  linear 
terms.  In  the  discussion  of  the  stability  of  a  solution,  it  maybe  justifiable 
to  neglect  all  except  the  linear  terms  when  the  differential  equations  are 
infinite  power  series;  but  with  these  restrictions,  which  are  inadmissible  in 
a  treatment  aiming  at  rigor,  it  is  not  possible  to  determine  whether  or  not 
periodic  solutions  exist.  Poincare  made  a  few  remarks*  upon  this  subject, 
relating  his  methods  to  the  equations  of  Hill,  which  lack  the  parallactic 
terms.  Burrau  discovered  several  orbits  in  a  special  case  from  successive 
trial  computations  by  mechanical  quadratures.!  Perchot  and  Mascart 
treated  the  special  case  in  which  the  finite  masses  are  equal. f  Sir  George 
Darwin  found  examples  of  these  orbits  about  two  of  the  points  of  equilib- 
rium in  his  celebrated  memoir  on  Periodic  Orbits. ||  His  methods,  like  those 
of  Burrau,  were  purely  numerical.  Under  the  assumption  that  the  orbits 
exist,  Plummer  gave  a  convenient  literal  development  of  expressions  for  the 
coordinates.!  His  method  is  simple,  but  apparently  it  is  not  easily  extens- 
ible to  most  of  the  more  complicated  cases.  All  of  the  writers  mentioned 
have  treated  the  problem  only  in  the  plane  of  motion  of  the  finite  masses. 
It  would  be  practically  impossible  to  discover  three-dimensional  orbits  by 
numerical  processes,  but  there  would  be  no  difficulty  in  applying  Plummer's 
method  to  infinitesimal  satellites  oscillating  in  three  dimensions  when  the 
finite  bodies  describe  circular  orbits. 

76.  The  Differential  Equations  of  Motion. — Let  us  take  the  origin  at 
the  center  of  gravity  of  the  system  and  refer  the  motion  of  the  infinitesimal 
body  to  a  set  of  axes,  £,  rj,  f .  We  will  choose  the  £  and  ij-axes  in  the  plane 
of  motion  of  the  finite  bodies,  and  suppose  that  they  rotate  in  the  direction 
of  motion  of  the  system,  with  the  same  angular  velocity.  The  initial  position 
of  the  axes  will  be  determined  so  that  the  finite  bodies  continually  lie  on 
the  £-axis.  The  distance  between  the  finite  bodies  will  be  taken  as  the 
unit  of  length,  the  sum  of  the  masses  as  the  unit  of  mass,  and  the  unit  of 
time  will  be  chosen  so  that  the  Gaussian  constant  is  unity.  Let  the  masses 

*Les  Melhodes  Nouvelles  de  la  Mecanique  Celeste,  vol.  I  (1892),  p.  159. 
\Aslronomiache  Nachrichlen,  Nos.  3230,  3251  (1894). 

\Bulletin  Aslronomique,  vol.  XII  (1895),  p.  329.     Apparently  their  work  is  vitiated  by  an  error  in  estab- 
lishing the  existence  of  the  solutions,  and  their  construction  fails  where  they  stopped. 
\\Acta  Mathematica,  vol.  XXI  (1897),  p.  99. 
^Monthly  Notices,  Royal  Astronomical  Society,  vol.  LXIII  (1903),  p.  436,  and  vol.  LXIV  (1903),  p.  98. 


OSCILLATING   SATELLITES  —  FIRST   METHOD. 


153 


of  the  finite  bodies  be  1-^  and  M-  The  units  have  been  chosen  so  that 
the  angular  velocity  of  revolution  is  unity.  The  differential  equations  of 
motion  of  the  infinitesimal  body  are  then* 


(1) 


Necessary  and  sufficient  conditions  for  a  solution  in  which  the  infini- 
tesimal body  is  at  rest  relatively  to  the  finite  masses  are 

dU_9U_dV_Q  m 

dt~  dr,-~d{- 

The  second  and  third  of  these  equations  are  satisfied  by  ij  =  f  =  0,  what- 
ever £  may  be.  The  first  equation  has  three  solutions:!  (a)  one  between 
+  oo  and  the  finite  mass  n,  (b)  one  between  n  and  1  —  n,  and  (c)  one  between 

1  —  n  and  —  oo . 


(C) 


(b)  JU    fa) 

— * — S » — 


Fio.  2. 


These  three  solutions  are  the  real  positive  roots  of  the  quintic  equations 
(6)         i\—  (3— p)rJ+(  3  —  2n)i\— nrl+2nrt— M  =  0,  (3) 

where  p  =  2  — r,,  and  where  r,  is  the  distance  from  n  to  the  equilibrium 
point.    The  real  positive  solutions  of  (3)  are  respectively 

(a) 

•  >  9f        ^  W  S  If        ^   19    S 

It* 

(4) 


2  -f- 


•Moulton'a  Introduction  to  Ctlettial  Medutniet,  p.  185. 

fSee  Introduction  to  CeUttial  Mechanict,  Art.  121,  and  especially  Charlier'i  Die  Mtchanik  de*  HimmeU, 
TO).  II,  pp.  102-111,  for  a  detailed  diacuaaion. 


154  PERIODIC   ORBITS. 

Suppose  the  coordinates  of  (a),  (6),  or  (c)  are  £=$0>  *7  =  0,  T=0,  the  value 
of  £0  depending  upon  which  point  is  in  question.  It  will  not  be  necessary 
to  distinguish  among  them  except  in  numerical  computation.  Now  give  the 
infinitesimal  body  a  small  displacement  from  one  of  these  points,  and  a 
small  velocity  with  respect  to  the  finite  masses  such  that 

h  z', 

(5) 


y,  r=+  z, 


- 
dt  "          dt  dt~         dt  dt  ~      ~  dt 

The  differential  equations  (1)  are  transformed  by  these  relations  into 


__,p(,     ,2     , 
dt  ~dx'~        l(    '  y  ' 


-  +2        -        -  v'P  (x'  y'2  z'2) 
df      '  dt  ~dy'~y     *(  •    >y  '      )j 


where  Pl  ,  P2,  and  P3  are  power  series  in  x',  y'\  and  z'2. 

77.  Regions  of  Convergence  of  the  Series  Pl  ,  P2  ,  P3  .  —  It  follows  from 
the  form  of  U  in  equations  (6)  that  Plt  Pt,  and  P3  converge  for  the  common 
region  of  convergence  of  the  expansions  of  \/rl  and  l/r2  .  We  are  consider- 
ing only  real  values  of  x',  y',  and  z',  and  consequently  the  conditions  for 
the  convergence  of  the  expansions  of  l/rt  and  l/r2  as  power  series  in  x',  y', 
and  z'  are  respectively 


_«        2x'       x  . 

*  2 

The  surfaces  which  bound  the  regions  of  convergence  of  the  expansions  are 
obtained  by  replacing  these  inequalities  by  equalities.  For  the  convergence 
of  the  expansion  of  \/rl  ,  the  equations  of  the  bounding  surfaces  are 

0,  i 

0.  j 

The  first  is  the  equation  of  the  point  occupied  by  the  finite  body  1  —  ju. 
The  second  is  the  equation  of  a  sphere  whose  center  is  at  1  —  M  and  whose 
radius  is  V2(|0+M)2-  The  convergence  of  the  expansion  of  l/r:  holds 
for  the  space  between  the  point  and  this  sphere. 


O8CILL.MIM.    >  \  I  1  II  1  I  IS —FIRST   METHOD.  155 

The  o(|iiations  of  corresponding  surfaces  for  the  expansion  of  1/r,  arc 

0,        1 
0. 

These  are  respectively  tin-  equations  of  the  point  occupied  )>\  the  mass 
H  and  of  a  sphere  whose  center  is  at  n  and  whose  radius  is  V2(£0—  1 +/*)'• 
The  convergence  of  the  expansion  of  1/r,  holds  for  all  points  within  this 
sphere  except  the  center. 

The  distances  from  I-M  and  n  to  the  point  (a)  are  respectively  &+M 
and  £„—  1  +/*.  The  radii  of  the  spheres  which  have  been  defined  in  (8)  and 
(9)  are  \/2  times  these  distances.  Since  \/2  (£9+n)-l>  \/2  (&— I+M),  the 
sphere  around  1  —  M  as  a  center  is  entirely  outside  of  the  one  around  M  as  a 
center.  (  'onseqiiently,  the  series  P, ,  P, ,  and  P,  converge  in  the  case  of  the 
t  must".. i -niation  to  the  point  (a)  for  all  points  within  the  sphere  whose  center  is 
at  n  and  whose  radius  is  \/2  (£O+M)>  except  the  point  M  itself. 

The  distances  from  1  -n  and  n  to  the  point  (6)  are  £0+M  and  %/({„— !+/*)*. 
The  radii  of  the  two  spheres  are  \/2~  times  these  distances,  and  hence  they 
both  include  the  point  (b)  in  their  interiors.  In  this  case  the  two  spheres 
intersect  unless  n  is  small,  when  one  will  be  entirely  within  the  other. 

The  distances  from  1  -^  and  n  to  the  point  (c)  are  —  £,— M  and  1  —  &— p. 
Since  -v/2  (1  -  £„  -  /*)  -  1  >  \/2  (  -£»  -  M),  the  sphere  around  n  as  a  center 
includes  in  its  interior  the  one  around  1  -/*  as  a  center.  The  latter  includes 
(c)  in  its  interior,  and  everywhere  within  it,  except  at  1  —  /*,  the  series  P, , 
P,,  and  P,  converge. 

78.  Introduction  of  the  Parameters  e  and  6 . — Let  us  now  make  the 
transformations 

x'  =  xt',         y'  =  yt',         z'  =  zt'        («VO),         *-/e=(l  +  5)r,      (10) 

where  «'  and  5  are  constant,  but  at  present  undetermined,  parameters. 
Then  equations  (6)  become 


1, 


(11) 


where  A'. ,  Y, ,  Z.  are  homogeneous  functions  of  x,  y,  and  z  of  degree  n. 
These  differential  equations  are  valid  for  all  values  of  x,  y,  z,  and  «'  satis- 
fying the  conditions  for  convergence  which  have  been  developed. 


156  PERIODIC   ORBITS. 

We  shall  now  generalize  the  parameter  e  (see  §13)  by  replacing  it 
everywhere  by  e,  where  e  may  have  the  value  zero  or  any  value  in  its 
neighborhood.  When  e  ^  e',  the  differential  equations  belong  to  a  purely 
mathematical  problem;  but  when  e  =  e'  they  belong  to  the  physical  problem. 
Since  the  value  of  e'  has  not  been  specified  except  that  it  is  distinct  from 
zero,  the  generalization  may  appear  trivial,  but  the  same  method  can  be 
used  where  the  parameter  corresponding  to  d  does  not  have  this  arbitrary 
character,  and  where  the  device  is  of  the  highest  importance.  We  have 
therefore  to  consider  the  differential  equations 


-  2(1  +  6)        = 


(12) 


79.  Jacobi's  Integral.  —  Equations  (1)  admit  the  integral 


where  C  is  the  constant  of  integration.  This  integral  was  first  given  by 
Jacobi  in  Comptes  Rendus  de  I'  Academic  des  Sciences  de  Paris,  vol.  Ill,  p.  59. 
For  equations  (12)  there  is  the  corresponding  integral 


where  U  is  now  a  power  series  in  e. 

80.  The  Symmetry  Theorem.  —  Let  us  consider  the  solution  of  equations 
(12)  and  suppose  that  at  r  =  0  we  have 

dy  _  dz  _  dx  _n 

:ClJ  dr~  dr~~  ~  dr~ 

that  is,  that  the  infinitesimal  body  crosses  the  .r-axis  perpendicularly  at 
T  =  0.     The  solution  will  have  the  form 


Now  transform  equations  (12)  by  the  substitution 

x=+x',  y=-y',  z=-z',         T=-T', 

dx  _   _  dx'  dy=i_dy^  dz  _    .  dz' 

dr  ~    ~  dr'  '  dr  =     h  dr'  '  dr       r  dr'  ' 


OSCILLATING   SATELLITES — FIRST  METHOD.  157 

The  equations  in  the  new  variables  ;m-  precisely  the  same  as  in  the  old; 
consequently,  if  the  values  of  the  dependent  variables  at  T'  =  0  are 


•-<••  -*»  -<»        "'-*'-  5? 

the  solution  is 


57=/;«,    g£-/;(o,    37  =/;(''). 

Now  it  follows  from  the  relations  between  the  two  sets  of  variables  that 

/,(T)  =  +/,(T')  =  +/,(  -  T),          /;<r)  =  -/((TO  -  -/;  (  -  T), 
/,(T)  =  -/,(r')  =  -/,(  -  r),  /I(r)  =  +/I(T')  =  +/i  (  -  r), 


Therefore,  if  the  infinitesimal  body  is  projected  perpendicularly  from  the 
x-axis,  :hen  x,  dy/dr,  and  dz/dr  are  even  functions  of  T,  and  dx/dr,  y, 
and  2  are  odd  functions  of  T;  that  is,  the  orbit  is  geometrically  symmetrical 
with  respect  to  the  x-axis,  and  it  is  symmetrical  in  T  with  respect  to  the 
time  of  crossing. 

81.  Outline  of  Steps  for  Proving  the  Existence  of  Periodic  Solutions 
of  Equations  (12).  —  In  (12)  we  put  6  =  e  =  0  and  find  the  general  solutions 
of  the  resulting  equations.  For  special  values  of  the  constants  of  inte- 
gration there  are  periodic  solutions.  Then  we  change  the  initial  values 
of  the  dependent  variables  by  small  amounts  and  take  5^0,  t^O.  The 
equations  are  integrated  as  power  series  in  6  and  e  and  in  the  increments 
to  the  initial  values  of  the  dependent  variables.  By  §11,  these  parameters 
can  be  taken  so  small  in  numerical  value  that  the  solutions  will  converge 
for  all  T  in  any  preassigned  range,  and  in  particular  for  the  periods  of 
the  periodic  solutions  obtained  when  «  =  0. 

After  having  formed  the  solutions  as  power  series  in  the  parameters, 
the  conditions  are  imposed  that  the  solutions  shall  be  periodic  with  the 
same  period  in  r(not  in  0  as  the  generating  solutions  have  for  e  =  0.  These 
conditions  are  that  the  orbit  shall  re-enter  at  the  end  of  the  period;  or,  in 
the  case  of  the  symmetrical  orbits,  that  they  shall  cross  the  x-axis  perpen- 
dicularly at  the  half  period.  These  periodicity  conditions  are  relations 
imposed  upon  the  initial  values  of  the  dependent  variables  and  upon  6. 
It  is  shown  that  these  conditions  can  be  satisfied  by  expressing  5  and  the 
initial  values  of  the  dependent  variables  as  power  series  in  «,  and  (hoe 
series  converge  for  the  modulus  of  t  sufficiently  small. 


158 


PERIODIC    ORBITS. 


82.  General  Solutions  of  Equations  (12)  for  8  =  e  =  0.  —  On  referring 
to  equations  (1)  and  the  succeeding  transformations,  we  find  that  equations 
(12),  for  6  =  e  =  0,  become  explicitly* 


(14) 


where 


J- 


The  third  equation  of  (14)  is  independent  of  the  first  two,  and  its 
general  solution  is 

z  =  c1cosVAr  +c^siiiVAT,  (16) 

where  ct  and  c2  are  the  constants  of  integration. 

The  first  two  equations  of  (14)  are  linear  and  homogeneous,  and  they 
have  constant  coefficients.     To  find  their  solution,  let 

x  =  Ke*r,  y  =  Le*T,  (17) 

where  K  and  L  are  constants.     The  conditions  that  these  expressions  shall 
identically  satisfy  (14)  are 

[X2-(1+2A)]#-2XL  =  0,  2X#+[X2-(1-A)]L  =  0.      (18) 

In  order  that  these  equations  may  have  a  solution  for  K  and  L  other  than 
the  trivial  one  K=L  =  0,  we  must  impose  the  condition 


A= 


,  -2X 

+2X,    X2-(l- 


(19) 


We  shall  now  discuss  the  roots  of  this  biquadratic  equation  in  X.     Its 
discriminant  is 

D  =  (2-A)*-4:(l-A)(l+2A)  =  (9A-8)A.  (20) 

We  shall  show  that  1—  A  is  negative  for  each  of  the  points  (a),  (b),  and  (c) 
for  all  values  of  /4>  and  therefore  that  D  is  positive.     From  (15)  we  have 


(21) 


At  the  points  (a),  (b),  and  (c)  we  have 


_          __ 
dx       dr,  dx      drt  dx 


*See  also  Introduction  to  Celestial  Mechanics,  and  Charlier's  Mechanik  des  Himmels,  vol.  II,  pp.  117-137. 
fFirst  proved  for  all  it  by  H.  C.  Plummer,  Monthly  Notices  of  Royal  Astronomical  Society,  vol.  LXII  (1901). 


OSCILLATING    SATELLITES  —  FIRST   METHOD.  159 

It  is  found  from  the  definition  of  U  in  (1)  that 


For  the  point  (a)  the  relation  1>H  \\ccn  r|0)  and  r™'  is  rjm  =  1+rJ",  and  therefore 

dr™  _dr™_ 
dx         dx 

Hence,  equations  (22)  and  (23)  give  for  this  point 


m       11  r<«       1 

.  -^J=  -M[T.  -^« 


Therefore,  since  the  first  two  factors  are  positive  while  the  third  is  negative, 
(21)  becomes,  for  the  equilibrium  point  (a), 

l-^r)<0.  (24) 

Similarly,  since  only  the  second  factor  in  the  expression  for  1—  A  is 
negative,  for  the  point  (6)  we  find 


<25) 

For  the  point  (c)  we  have  the  corresponding  equations 

(26) 


Then  1—^1  is  negative  because  the  third  factor  alone  is  negative. 

Since  1—  A  is  negative  in  every  case  for  O^M^O.5,  it  follows  that  two 
of  the  roots  of  (19)  are  real  and  equal  numerically  but  opposite  in  sign, 
and  that  the  other  two  are  conjugate  pure  imaginaries.  Let  the  real  roots 
be  ±p  and  the  imaginary  ±<rV—  1.  For  each  of  these  roots  there  is  a 
particular  solution  (17),  and  the  general  solution  is 


x    f\.l  _          ,  _-,  -  .  — ,  -     .  — ,        ,  . 

y  =  LI  e"     T+  Lj  e~*     T-\-  Lt  tf" -\-  L4  e~pT,  J 

where,  from  (18), 

(28) 


7  I       P—      —  V  ,   Tf          —t  r 

L,=  —  —  -Kt=        fmA,=  -^—  L4. 

The  constants  m  and  n  are  defined  by  these  equations. 


160  PERIODIC    ORBITS. 

83.  Periodic  Solutions  when  d  =  e  =  0.  —  The  general  solution  of  equations 
(14)  is  contained  in  equations  (16)  and  (27).     One  periodic  solution  is 


the  period  of  this  solution  being  27T/VZ.  The  constants  ct  and  c2  ,  and 
the  ta  on  which  T  depends,  are  not  independent.  We  shall  suppose  t0  is 
taken  so  that  Cj  =  0,  c2  =  c/VA.  Then  one  of  the  periodic  generating 
solutions  which  we  have  to  consider  is 

(29) 


These  will  be  called  Orbits  of  Class  A. 

Another  periodic  solution  of  the  differential  equations  (14)  is 


0  =  0. 

If  the  initial  conditions  are  real,  as  we  suppose,  then  /^  and  K2  are  conjugate 
complex  quantities.  We  shall  suppose  ta  is  chosen  so  that  the  imaginary 
part  of  KI  and  Kt  is  zero.  Let  a/2  represent  their  real  part.  Then  we 
have,  as  the  second  periodic  generating  solution, 

£  =  acosor,  y=  —  nasinor,  z  =  0,  (30) 

the  period  being  2w/ff.     These  will  be  called  Orbits  of  Class  B. 

A  third  periodic  solution  will  exist  if  a  and  V~A  are  commensurable. 
We  shall  first  prove  the  possibility  of  their  being  commensurable.  The 
condition  for  commensurability  is  <r/VA  =  p/q,  where  p  and  q  are  positive 
integers.  Since  0V  —  1  satisfies  (19),  we  have  from  this  relation 


The  solution  of  this  quadratic  equation  for  A  is  found  to  be 


A= 


In  order  to  establish  the  possibilitj^  of  the  commensurability  of  a-  and  VA,  it 
is  sufficient  to  show  that  p  and  q  can  be  assigned  such  positive  integral 
values  that  the  A  defined  in  (31)  shall  have  a  value  equal  to  that  obtained 
from  (15)  for  some  n  between  0  and  0.5. 

It  is  to  be  observed  first  that  the  solutions  of  (3)  are  continuous  func- 
tions of  M,  and  consequently  A,  as  defined  by  (15),  is  a  continuous  function 
of  p.  Therefore,  if  there  are  positive  integral  values  of  p  and  q  such  that 
the  A  defined  by  (31)  lies  in  the  range  of  values  of  A  as  defined  by  (15),  there 
are  infinitely  many  values  of  ^  for  which  cr  and  VA  are  commensurable. 


OSCILLATING    SATELLITES  —  FIRST   METHOD.  161 

On  taking  the  upper  sign  in  (31),  we  find  that  in  order  that  A  may  be 
real  and  positive  \vc  inu-t  have  1  <p'/qt<9/S.  For  values  of  j>  and  q  satis- 
fying these  inequalities.  .1  lies  between  +00  and  8/5.  On  taking  the  lower 
sign  in  ('.\\  ).  we  must  have  0<pt/qt  <  9/8  in  order  that  A  may  be  real  and 
positive.  The  values  of  .4  for  these  limits  are  unity  and  8/5.  Equation 
(31)  takes  the  indeterminate  form  O-v-0  for  p  =  q,  but  it  is  easily  found 
that  the  corresponding  value  of  A  is  unity.  For  0<pt/qt<  1,  the  value  of 
.1  .  defined  by  (31)  with  the  lower  sign,  is  less  than  unity.  On  taking  both 
the  upper  and  lower  signs,  it  follows  that  A  takes  values  in  every  finite 
interval  from  1  to  oo  as  p'/g1  goes  over  all  rational  fractions  from  0  to  9/8. 

It  was  proved  in  the  preceding  article  that  1—  A<0  for  each  one  of 
the  three  solution  points  (a),  (6),  and  (c).  Therefore  for  these  points  and 
0^/*<0.5,  we  have  A  >  1.  Consequently,  there  arc  infinitely  many  values 
of  n  between  0  and  0.5,  such  that  a  and  VA  are  commensurable.  \Yhen 
the  commensurability  relation  is  satisfied  we  have  the  periodic  solution 


z  =  c,cosvTr+c18inv'jT. 

We  can  choose  t,  so  that  c,  =  0,  and  let  a  represent  twice  the  real  part  of  A', 
and  A',,  and  b  twice  the  imaginary  part  of  —  A\  and  Kt  .  Then  this  solution 
becomes 

y  =  nas\n  at  +n&cosar,     z=  —r-   sinVAr.      (32) 


The  period  of  this  solution  is  P  =  2irp/ff  =  2irq/VA.  In  this  period  x  and 
y  make  p  complete  oscillations,  and  z  makes  q  complete  oscillations.  These 
will  be  called  Orbits  of  Class  C. 

84.  Normal  Form  for  the  Differential  Equations.  —  We  are  about  to 
prove  that  when  e^O  the  initial  values  of  x,  y,  z,  and  their  derivatives 
can  be  so  determined,  depending  on  «,  that  periodic  solutions  having  the 
periods  of  (29)  and  (30)  exist  for  all  values  of  e  sufficiently  small,  and  reduce 
to  these  solutions  for  <=0.  In  this  discussion  it  is  convenient  to  have  the 
differential  equations  in  a  normal  form,  and  it  is  necessary  to  compute  the 
first  terms  of  the  solutions  as  power  series  in  5,  «,  and  the  increments  to 
the  initial  values  of  the  dependent  variables. 

The  linear  terms  of  (12)  are  found  by  (14)  to  be 


162  PERIODIC    ORBITS. 

The  general  solution  of  the  first  two  of  these  equations  is 


2/  =  nv/-l  [K, 
Therefore  we  see  that  the  transformation 

X   = 


(33) 


changes  (12)  into 


,«N,  - 

(  2(m(r-np)         h 

+n(l  +  8)[]e 
(          w4-  -  h 


(34) 


where 


Ft  -    +  i-**  +  JL  ,  n  -  1~M  -l-  JL  , 

-O     —    +      r<OM     —  r<0)4  '  '          (0)5     ~T  r(0)5 

—      '1  —  '1  'l  '2 


(35) 


In  5  the  upper,  middle,  or  lower  signs  are  to  be  used  according  as  solutions 
in  the  vicinity  of  the  point  (a),  (b),  or  (c)  are  being  treated.  This  trans- 
formation is  always  valid,  since  we  find  from  (28)  that 

(36) 


<rp 

It  is  not  advantageous  to  transform  the  z-equation,  nor  [  ]  and  \  [  . 

It  follows  from  (1)  that  whenever  the  infinitesimal  body  is  displaced 
from  the  xy-plane,  it  is  always  subject  to  a  component  of  acceleration  toward 
this  plane.  Therefore  it  can  not  revolve  in  a  closed  orbit  entirely  on  one 
side  of  the  xy-plaue.  Hence  we  may  determine  £0  so  that  z  =  0  when  r  =  0. 
That  is,  without  loss  of  generality  we  can  take  the  initial  value  of  z  as  zero. 


OSCILLATING    8ATKI.U  I  KS      KIKST    MKTHOD.  163 

Let  the  initial  conditions  be 

i/(  =  «,+a,          (i-l,...,4),  z  =  0,          rf7=c+'y' 

when-  the  nt  and  c  can  be  given  such  values  that  we  shall  have,  for  e  =  0, 
either  (29),  (30),  or  (32).  The  a,  and  y  are  to  be  determined  in  terms  of 
e  so  that  the  solutions  shall  remain  periodic  for  e^O. 

Instead  of  integrating  (34)  directly  in  powers  of  all  the  parameters 
a  .....  ,  a4  ,  -y,  5,  and  e,  we  can  more  conveniently  integrate  them  as 
power  series  in  e;  the  parameter  5  can  be  introduced  in  connection  with  T, 
since  it  always  occurs  in  the  combination  (l  +  6)r;  and  the  parameters 
a,,  .  .  .  ,  a4,  and  y  can  be  introduced  when  the  constants  of  integration  are 
determined. 

The  terms  which  are  independent  of  e  are  defined  by  the  equations 


(37) 


The  solutions  of  these  equations  which  satisfy  the  initial  conditions  are 


u™  =  (a4+a4) 


(38) 


These  expressions  can  at  once  be  expanded  as  power  series  in  5.  The 
coefficients  of  higher  powers  of  t  can  be  found  by  the  usual  process,  but  we 
shall  not  need  them  in  proving  the  existence  of  periodic  solutions. 

85.  Existence  of  Periodic  Orbits  of  Class  A.—  For  e  =  0  the  coordinates 
in  these  orbits  are  given  in  (29).  Therefore,  since  the  determinant  of  the 
transformation  (33),  viz.,  A  =  4(mp  +  7i<r)  (wa-np)v^T,  is  distinct  from 
zero,  it  follows  that  in  this  case  a,  =  o1  =  a,=a4  =  0.  The  general  solutions 
of  (34)  are  of  the  form 

ul  =  P,(al  ,  .  .  .  ,  o4  ,  y,  5,  e;  T)         (i  =  l,  .  .  .  ,  4), 
z=P,(a,,  .  .  .  ,at,y,  6,  «;  T),  (39) 

z'  =  P,(a,  ,  .  .  .  ,  a4,  y,  &,  c,  T), 

where  the  P«  are  power  series  in  a,  ,  .  .  .  ,  a4  ,  y,  d,  and  «,  and  where  z' 
denotes  the  derivative  of  z  with  respect  to  T. 

Since  equations  (34)  do  not  involve  T  explicitly,  sufficient  conditions 
that  the  solutions  (39)  shall  be  periodic  in  T  with  the  period  2r/\fA  are 

,  ...,«,,  t,  «,  •  -p'(a»  '  '  '  '  a"  ^a'''0)=0-          (40) 


164  PERIODIC    ORBITS. 

These  conditions  are  not  all  necessary,  for  it  can  be  shown  that  the  last 
one  is  a  consequence  of  the  first  five.     If  we  make  the  transformation 


ut  =  at+vt,        2 
the  integral  (13)  may  be  written 
F(at+vt  ,  *±2  sinVJr+f  ,  (c+7)cosVlT+r',  5,  e)  -F(o(>  0,  c+j,  d,  e)  =0.    (41) 

This  equation  is  satisfied  at  r  =  2ir/VA  by  vt  =  £=  f'=  0,  and  we  find  from 
the  explicit  form  of  F  in  (13)  that  for  these  values 


=:  -i 

A-l]+tQ3, 

-v=  2L  _ 

=  0,  [e 


0= 


It  follows  that  (41)  can  be  solved  for  f'  as  a  power  series  in  a,,  7,  5, 
e,  !>!,...,  t>4  ,  f,  which  vanishes  with  i>4  =  f  =  0.  That  is,  if  u1}  .  .  .  ,  w4,  and 
z  retake  their  initial  values  at  r  =  2ir/VA,  z'  also  retakes  its  initial  value. 
Hence  we  can  suppress  the  last  equation  of  (40)  and  consider  the  solution 
of  the  first  five  equations. 

It  follows  from  (38)  that  the  explicit  forms  of  the  first  terms  of  (40)  are 


(42) 


where  the  Q4  are  power  series  in  the  cif,  j,  8,  and  e.  The  coefficients  of 
a3  and  a4  are  always  distinct  from  zero,  and  the  parts  of  the  coefficients 
of  at  and  a2  which  are  independent  of  5  vanish  only  if  o/VZ  is  an  integer. 
We  shall  suppose  at  present  that  this  ratio  is  not  an  integer,  and  that  it  is 
incommensurable.  Part  of  the  discussion  becomes  quite  different  when  it  is 
commensurable,  and  this  case  will  be  taken  up  when  we  discuss,  in  §96, 
the  question  of  the  existence  of  orbits  of  Class  C. 

It  follows  from  (34)  that  since  [  ]  and  {  \  involve  terms  in  z2  alone, 
and  since  za  does  not  vanish  identically  for  at  =  y  =  5  =  0,  the  Qt  carry  terms 
in  e  alone.  The  determinant  of  the  linear  terms  in  an  .  .  .  ,  a4  of  the 
first  four  equations  of  (42)  is  the  product  of  the  coefficients  of  04  ,  .  .  .  ,  a4 , 
and  is  distinct  from  zero.  Therefore  these'  equations  can  be  solved  for 
a:  ,  .  .  .  ,  a4  in  the  form 

a,  =  6^(7,8,6),  (43) 

where  the  Rt  are  power  series  in  7,  5,  and  e.  When  these  results  are 
substituted  in  the  last  equation  of  (42),  we  have 

=  e  P(y,  d,  e).  (44) 


OSCILLATING    SATKI.I.I  I  l-:s      MUST   METHOD.  165 

The  solution  of  this  equation  gives  us  the  periodic  orbits  in  question.  We 
have  the  two  arbitrary  parameters  7  and  6,  and  we  shall  show  first  that 
•  •an  not  give  6  an  arbitrary  value  and  solve  the  equation  for  7  as  a 
power  series  in  f,  vanishing  with  «. 

Suppose  that  <5  is  neither  /ero  nor  an  integer.  Then  equation  (44)  is  not 
satisfied  by  7  =  4  =  0,  and  the  solution  can  not  be  made.  Now  suppose  5  is 
zero  or  an  integer.  Then  the  left  member  of  (44)  vanishes,  and  the  equation 
is  divisible  by  t.  It  is,  in  fact,  divisible  by  «*.  It  is  seen  from  (34)  that  x 
and  y  do  not  enter  in  the  last  equation  except  in  terms  involving  «  as  a 
factor,  and  since  the  a,  defined  in  (43)  enter  only  through  x  and  y,  the 
part  of  (P  coining  from  the  first  four  equations  is  divi.-ible  by  e*.  It  is 
seen  also  that  the  part  of  the  right  member  of  the  last  equation  of  (34) 
which  is  independent  of  x  and  y  is  multiplied  by  «*.  Therefore  the  right 
member  of  (44)  is  divisible  by  e2.  We  shall  now  prove  that  after  e*  has 
been  divided  out  there  is  left  a  term  which  is  independent  of  7  and  «,  and 
which  is  distinct  from  zero. 

Terms  in  i-  in  the  right  member  of  (44)  are  introduced  both  through 
the  a,  defined  in  (43),  and  directly  in  the  integration  of  the  last  equation  of 
(34).  The  terms  obtained  in  the  former  way  involve  B  as  a  factor  and 
depend  upon  a  and  p;  the  terms  entering  in  the  latter  way  carry  C  as  a 
factor.  Hence,  if  the  coefficient  of  «'  is  to  be  identically  zero,  the  parts 
involving  B  and  C  as  factors  separately  must  be  zero.  We  shall  verify 
that  the  part  involving  C  as  a  factor  is  distinct  from  zero. 

The  coefficient  of  e1  in  (44),  so  far  as  it  is  independent  of  B  and  7,  is 
denned  by  the  equation 


(46) 
The  solution  of  this  equation  satisfying  the  conditions  2,  =  0,  z,'  =  0,  at  r  =  0,  is 

3«  (46) 


Consequently  the  last  equation  of  (42)  becomes 

-0-  (47) 


Hence,  after  division  by  e1,  there  is  a  term  independent  of  both  7  and  «, 
and  the  solution  for  7  as  a  power  series  in  «,  vanishing  with  «,  does  not  exist. 
That  is,  periodic  solutions  of  the  type  in  question  do  not  exist. 


166  PERIODIC    ORBITS. 

Now  let  us  give  7  an  arbitrary  value  and  attempt  to  solve  equations 
(42)  for  d!  ,  .  .  .  ,  a4 ,  and  5  as  power  series  in  e,  vanishing  with  e.  Since 
c  is  arbitrary,  we  may  put  7  equal  to  zero  without  loss  of  generality.  Or, 
more  conveniently  for  the  construction  of  the  periodic  solution,  we  may 
give  7  such  a  value  that  z'  =  c/VZ  for  T  =  0,  whatever  e  may  be.  But  for 
simplicity  in  writing  we  shall  suppose  that  7  is  included  in  c.  The  first  four 
equations  can  be  solved  for  a, ,  .  .  .  ,  a4  in  terms  of  5  and  e,  and  the 
results  substituted  in  the  last.  The  result  differs  from  that  above  only  in 
the  terms  multiplied  by  e ,  and,  as  before,  we  find  that  the  lowest  term  in  e 
alone  has  e2  as  a  factor.  There  is  a  linear  term  in  d  alone  whose  coefficient 
is  2-irc/VA.  Therefore,  after  a, ,  .  .  .  ,  a4  have  been  eliminated  by  means 
of  the  first  four  equations,  the  last  equation  can  be  solved  for  5  as  power 
series  in  e,  the  term  of  lowest  degree  being  e2.  When  this  result  is  substituted 
in  the  solutions  of  the  first  four  equations,  we  have  ot ,  .  .  .  ,  a4  expressed 
as  power  series  in  e,  vanishing  with  e.  That  is,  when  7  =  0  the  solutions 
of  (42)  have  the  form 

5  =  €2p(e),  af  =  ep,(e)  (t  =  l,  .  .  .  ,  4),     (48) 

where  p  and  the  pf  are  power  series  in  e.  When  these  results  are  substituted 
in  (39),  we  have 

z  =  z0+e2g(e;  T),        ui  =  tqi(r,T)  ({  =  !,...,  4),     (49) 

where  q  and  the  qt  are  power  series  in  e  and  are  periodic  in  T  with  the 
period  2-ir/VA.  These  series  converge  for  kl  sufficiently  small.  The  circle 
of  convergence  is  determined  by  the  singularities  which  are  present  in  the 
differential  equations  (34),  which  are  introduced  in  forming  (39),  and  which 
are  introduced  in  the  solution  of  (42).  Since  the  right  members  of  (49) 
converge  and  are  periodic  for  all  |  e  \  sufficiently  small,  the  coefficient  of  each 
power  of  e  separately  is  periodic. 

86.  Some  Properties  of  Solutions  of  Class  A. — It  will  now  be  shown 
that  the  orbits  under  consideration  are  re-entrant  after  one  revolution,  and 
that  they  cross  the  x-axis  perpendicularly. 

Let  us  find  the  orbits  whose  periods  are  2,v^^/^/rA,  v  being  an  integer. 
We  form  equations  analogous  to  (42)  simply  by  replacing  2  IT  by  2i>ir.  The 
determinant  of  the  linear  terms  in  e^  ,  .  .  .  ,  a4  is  distinct  from  zero  unless 
vv/\fA  is  an  integer.  We  exclude  this  case  here  and  treat  it  when  we  con- 
sider orbits  of  Class  C.  Therefore  the  first  four  equations  can  be  solved  for 
aL ,  .  .  .  ,  a4  as  power  series  in  7,  d,  and  e.  On  substituting  the  results  in 
the  last  equation,  we  find,  as  before,  that  the  solution  can  not  be  made  for 
7,  taking  5  arbitrary,  but  that  the  solution  for  6  as  a  power  series  in  e  is 
unique.  That  is,  the  solution  for  at  ,  .  .  .  ,  a4 ,  and  5  as  power  series  in  e, 
vanishing  with  e,  is  unique.  Hence  for  a  given  value  of  e  there  is  a 
single  orbit  of  Class  A  having  the  period  2i>ir/VA.  We  have  shown  also 


OSCILLATING    SATELLITES— FIKST   METHOD.  167 

that  for  a  given  value  of  t  there  is  one  orbit  of  Class  A  having  the  period 
2T/\~A.  Since  an  orbit  of  period  2w/VA  has  also  the  period  2.vr/\fAt 
the  latin-  are  included  in  the  former.  It  follows,  therefore,  from  the  unique- 
ness of  both  orbits  for  a  given  «,  and  from  the  fact  that,  for  «  =  0,  they 
re-enter  after  the  period  2T/V/71",  that  all  orbits  of  this  class  re-enter  after 
a  single  revolution. 

Let  us  now  suppose  that,  at  T  =  0,  we  have  dx/dr  =  j/  =  z  =  0;  that  is, 
that  the  orbit  crosses  the  x-axis  perpendicularly  at  r  =  0.  It  follows  from 
equation  (33)  that 

i«l(0)-«1(0)  =  0>     tt,(0)-t«4(0)  =  0;  (50) 

whence 

fll  =  °l   >  «1  =  fl4   • 

The  e< [nations  corresponding  to  (39)  will  now  be  power  series  in  o, ,  a, ,  y,  6, 
and  t.  \Ye  may  suppose  -y  =  0  on  the  start,  for  orbits  that  may  be  found 
in  this  way  will  be  included  in  those  found  from  more  general  initial  condi- 
tions, under  which  it  was  always  permissible  to  put  y  equal  to  zero.  The 
orbits  obtained  with  these  initial  conditions  will  be  symmetrical  with  respect 
to  the  x-axis.  Therefore  necessary  and  sufficient  conditions  for  periodicity 
are  that  the  infinitesimal  body  shall  cross  the  z-axis  perpendicularly  at  the 
half  period.  These  conditions  are  that  at  r  =  T/  VA 

*-,-<-o. 

It  follows  from  (33)  that  these  conditions  imply  that,  at  T  =  T/VJ, 

M,— w,  =  0,       «,— M4  =  0,       2  =  0.  (51) 

These  conditions  give  us,  in  place  of  (42),  the  equations 


(52) 


where  Q( ,  Q'3,  and  Q'&  are  power  series  in  a, ,  a, ,  8,  and  «.  It  is  easy  to 
see  that  these  equations  are  solvable  uniquely  for  a, ,  a, ,  and  d  as  power 
series  in  «,  vanishing  with  t.  Therefore,  for  a  given  value  of  «  there  is  one, 
and  but  one,  of  these  symmetrical  periodic  orbits  of  this  class.  Since  for  a 
given  value  of  e  there  is  but  one  periodic  orbit  for  unrestricted  initial  con- 
ditions, it  follows  that  all  orbits  of  Class  A  cross  the  x-axis  perpendicularly  at 
every  half  period. 


" 


=  a,e  -« 

0=  sin2T(l  +6)  + 


168  PERIODIC    ORBITS. 

87.  Direct  Construction  of  the  Solutions  for  Class  A.  —  In  the  practical 
construction  of  the  solutions  for  the  orbits  of  Class  A  it  is  most  convenient 
to  use  equations  (12).  The  explicit  values  of  the  right  members  are 


(53) 


the  A,  B,  and  C  being  constants  which  are  defined  in  (15)  and  (35). 
The  x,  y,  z,  and  5  can  be  expanded  uniquely  as  series  of  the  form 


X1=(l+2A)x,    X2=£[ 

Y2=3Bxy, 
Z2=3Bxz, 


z=  -sinVJ  r  +  V^W,    «  =       «.«',     (54) 


where  the  Z<(T),  y((r),  and  zt(r)  are  each  periodic  with  the  period 
On  substituting  these  expressions  in  (12)  and  making  use  of  (53),  we  obtain 
a  series  of  sets  of  equations  for  the  determination  of  the  xt,  yt,  zt  ,  6,  . 
The  coefficients  of  e  in  (54)  must  satisfy  the  equations 


2^  -(1  +  2.1)*,=  \B  [-2xl+y20+z20], 


(55) 


But  x0  =  y0  =  0,  zc  =  c/VT  sin  y/~Ar.  Therefore  the  solution  of  (55)  which 
satisfies  the  conditions  that  x1  ,  ?/,  ,  and  z1  shall  be  periodic  with  the  period 
2ir/Vl,  and  that  z  =  0,  z'  =  c,  at  r  =  0,  whence  zl  (0)  =  z[  (0)  =  0,  is 


(56) 


9  /T 

z>  = 

Since  in  all  cases  A>1,  the  coefficients  are  always  finite. 

The  coefficients  of  e2  in  (54)  must  satisfy  the  differential  equations 


(57) 


Upon  substituting  the  values  of  z0  and  xl  ,  the  third  equation  becomes 


,  .    o   /J 

3  2  =    S1 


OSCILLATING    SATELLITES  —  FIRST   METHOD.  169 

The  solution  of  this  equation  will  not  be  periodic  imli-<  we  impose  the  con- 
dition that  the  coefficient  of  sinx.tr  shall  vanish.  This  condition  is 
satisfied  l>v  <•-(},  hut  this  leads  to  the  trivial  solution  z=t/=z  =  0.  If  we 
reject  this  solution,  we  may  use  the  condition  for  the  determination  of  5,. 
After  i(  has  been  satisfied,  the  periodic  solution  of  (57),  having  the  period 
2ir/v'7f  and  fulfilling  the  conditions  that  Zj  =  z't=0  at  r  =  0,  IS 


r=»=n  -27  £'(1-34  +  144*)  c*         9Cc| 

"*-  ' 


(fig) 


In  this  manner  the  construction  of  the  periodic  solution  can  be  continued 
as  far  as  may  be  desired.  We  shall  prove  this  statement  by  induction,  and 
at  the  same  time  we  shall  derive  certain  general  properties  of  the  solution 
which  are  sati.-fied  by  the  terms  already  computed. 

suppose  xt,  .  .  .  ,  £»_! ;  i/0 ,  .  .  .  ,  yn-\ ',  z0 ,  .  .  .  ,  2,_i ;  o, ,  .  .  .  ,  Sm-\ 
have  been  determined  and  that  they  have  the  following  properties: 

1.  The  xtj  and  ytj  are  identically  zero,  j  an  integer. 

2.  The  z,;+l  are  identically  zero,  j  an  integer. 

3.  The  function  x,/+1  is  a  sum  of  cosines  of  even  multiples  of  \f~Ar,  and 

the  highest  multiple  is  2j+2. 

4.  The  function  ytj+l  is  a  sum  of  sines  of  even  multiples  of  V~AT,  and 

the  highest  multiple  is  2j+2. 

5.  The  function  2,,  is  a  sum  of  sines  of  odd  multiples  of  vTr,  and 

the  highest  multiple  is  2j  +  l. 

6.  The  $,,+,  are  zero. 

It  will  now  be  shown  that  these  properties  hold  for  x.,yu,zu,  and  5. . 
The  terms  z. ,  y, ,  z* ,  and  5,  satisfy  the  differential  equations 


(59) 


-  2A  z,  «.+/?.(*„  y,,  zt,  *,), 


where  P. ,  Q»,  and  /?„  are  polynomials  in  x, ,  .  .  .  ,  &,  (j  =  l,  .  .  .  ,  n-1). 

It  is  seen  from  (12)  that  P.  and  Q»  involve  y,  and  x',  only  in  the  products 
y't&n-j  and  x',tn-,.  If  n  is  even,  these  terms  are  zero  by  properties  1  and  6,  for 
then  either  j  must  be  even  or  n—j  must  be  odd.  But  if  n  is  odd,  they  are 
in  general  not  zero. 


170  PERIODIC    ORBITS. 

We  shall  now  prove  that  Pn=0  if  n  is  even.  The  general  term  of  Pn  is 

rr  _  r"i   .        .    -^  .  7X                  »X'  .  /"  X'"  .  sir.   XJ2                   /«n\ 

ln~            y        »  d"  di"  ' 


where  Xj  ,  .  .  .  ,  XK  ,  .  .  .  ,  \"»  ;  MI  ,  •  •  •  ,  M?"  ;   Pi  ,  P2  ,  ft  ,  and  g2  are  all 
integers.     Since  the  5,  enter  only  through  (1  +  5)2,  the  exponents  ql  and  q2 

satisfy  the  relation 

2.  (61) 


The  exponents  and  subscripts  of  (60)  satisfy  the  following  relations  : 

(a)  /*!+••'  +AV  is  an  even  integer  because  the  right  member  of  the 

first  equation  of  (12)  is  a  function  of  yz. 

(b)  /*i'+  '  '  '  +MK"  is  an  even  integer  for  a  similar  reason. 

(c)  \!  ,  .  .  .  ,  XK  ,  X(  ,  .  .  .  ,  XK-  are  odd  integers  by  property  1. 

(d)  Xf  ,  .  .  .  |  XK»  are  even  integers  by  property  2. 

(e)  pl  and  p2  are  even  integers  by  property  6. 

(f)  MA+  •  •  •  +  M-c 


because  the  term  of  degree  MI+  •  •  •  +M«+M[+  •  • 
in  x,  ?/,  and  z  has  e  as  a  factor  to  a  degree  one  less  than  this  sum, 
the  terms  x^J,  .  .  .  introduce  e  to  the  degree  MiX1;  .  .  .  ,  and  the  sum 
of  the  exponents  of  e  must  equal  n. 

There  are  two  sub-cases,  according  as  MJ+  •  •  •  +M«  is  an  even  integer 
or  an  odd  integer.  When  ^4-  •  •  •  +MK  is  an  even  integer,  the  following 
statements  are  true: 

(a)    There  is  an  even  number  of  odd  /^  ,  .  .  .  ,  V,K. 

(/3)    MJ  Xt+  •  •  •  +MK  \c  is  an  even  integer  by  (c)  and  (a). 

(T)    Mi^i+  '  '  '  +/vX«'  is  an  even  integer  by  (a)  and  (c). 

(5)    juiX'+  •  •  •  +A&XK"  is  an  even  integer  by  (d). 

(«)    Pi  tfi+Pa  &  is  an  even  integer  by  (e). 

It  follows  from  the  assumption  that  MI+  •  •  •  +MK  is  even,  and  from  (a),  (b), 
(a)  ,  .  .  .  ,  (e)  that  the  left  member  of  (f  )  is  odd.  Therefore  in  this  case  Tn  is 
identically  zero  if  n  is  even,  and  in  general  is  not  identically  zero  if  n  is  odd. 

Suppose  now  that  jut+  •  •  •  +M*  is  an  odd  integer.     Then 

(a')    There  is  an  odd  number  of  odd  MI  ,  •  •  •  ,  M*  • 

(/3')    MI  ^1+  '  '  '  +M*  \  is  an  odd  integer  by  (c)  and  (a'). 
The  properties  (yf),  (d'),  and  («')  are  the  same  as  (7),  (d),  and  (e)  respectively. 
Therefore  the  left  member  of  (f)  is  again  odd,  and  hence  every  Tn  is  identi- 
cally zero  if  n  is  even,  and  in  general  not  identically  zero  if  n  is  odd.    It  follows 
that  Pn  is  identically  zero  if  n  is  even,  and  in  general  is  not  zero  if  n  is  odd. 

The  treatment  of  the  general  term  of  Qn  can  be  made  in  a  similar  way. 
The  only  differences  are  that  in  (a)  and  (7)  the  sums  are  individually  odd 
instead  of  even.  But  since  (f  )  involves  their  sum,  the  result  is  that  Qn  is 
identically  zero  if  n  is  even,  and  in  general  is  not  zero  if  n  is  odd. 


-i-ILLATINC    SATKI.I.IIK-       HUM     MKTIIOD.  171 

The  nen.-ral  term  in  ltn  has  the  form  of  »i()i.  where  tin-  subscripts  and 
(•x]M)iiciits  satisfy  the  relations: 

(a*)    //  +  '  '  '  +  /v  is  a»  even  integer  because  the  right  member  of 

the  third  equation  of    li'    is  a  function  of  y*. 
(b*)    n*  +  •  •  •  +»*••  is  an  odd  integer  because  the  ri»ht  member  of 

this  equation  involves  only  odd  powers  of  z. 
(c*),  (d*),  (e*),  and  (P)  are  the  same  as  (c),  (d),  (e),  and  (f)  respectively. 

Suppose  /i,+  •  •  •  +A*«  is  an  even  integer.  Then  (a*),  (0*),  (/),  (*")» 
and  (e*)  are  the  same  as  (a),  (/3),  (7),  (6),  and  («)  respectively.  Therefore  in 
t  his  case  t  he  left  member  of  (P)  is  an  even  integer.  .  It  is  shown  similarly  that 
the  same  result  is  true  when  /!,+  •••  +/*«  is  an  odd  integer.  Therefore,  #„ 
IN  identically  zero  if  n  /s  odd,  and  in  general  is  not  identically  zero  if  n  is  even. 

The  discussion  now  naturally  divides  into  two  cases,  viz.,  where  n  is 
even,  and  where  n  is  odd.  We  shall  treat  them  separately. 

Case  I.  We  shall  prove  that  if  n  is  even,  Rn  is  a  sum  of  sines  of  odd 
multiples  of  VAT,  the  highest  multiple  being  n+1.  Consider  the  general 
term  (60).  The  x^  are  all  cosines  of  even  multiples  of  VAT;  therefore 

the  product  x"1  •  •  •  x"'  is  a  sum  of  cosines  of  even  multiples.     Because 
\  x« 

of  property  3  and  the  properties  of  the  products  of  cosines  of  multiples  of 
an  argument,  it  follows  that  the  highest  multiple  which  occurs  is 


«..        (62) 

Similarly,  from  properties  4  and  (a*)  ,  it  follows  that  y"^.  •  •  •   y^f  is  a 
sum  of  cosines  of  even  multiples  of  VAT,  the  highest  multiple  being 

Mi(x;+i)+  •  •  •  +/V(X:<+I)=M;X;+  •  •  •  +M;X+M;+  •  •  •  +/*.  (63) 

From  properties  5  and  (b*),  it  follows  that/1,  •  •  •  £"  is  a  sum  of  sines 

*?  «" 

of  odd  multiples  of  VAT,  the  highest  multiple  being 

'-.  (64) 


On  taking  the  product  of  these  three  sets  of  terms,  we  find  that  Rn  is  a  sum 
of  sines  of  odd  multiples  of  VAT,  the  highest  multiple  being 


By  (P),  which  is  of  the  same  form  as  (f),  we  have 


For  those  terms  in  which  9,  =  9,  =  0,  we  have,  as  the  largest  value  of  N, 

tf-n+1.  (66) 


172  PERIODIC    ORBITS. 

Hence,  for  n  even,  equations  (59)  become 


(67) 


+C™+1sin(2j+l)VAT+  • 

where  the  ££+,  are  known  constants  which  depend  upon  the  coefficients  of 
the  terms  with  lower  subscripts.  The  solution  of  these  equations  which 
satisfies  the  periodicity  conditions  and  zn  =  z'a  =  Q  at  r  =  0,  is 

(7<n) 
xn  =  n  =  Q  8n= 


(n)      _ 


f,-=1  -} 

'2/' 


n/2 


(68) 


Case  II.  We  shall  prove  that,  if  n  is  odd,  Pn  is  a  sum  of  cosines  of 
even  multiples  of  VAT,  the  highest  multiple  being  n+1.  From  properties 

1  and  3  it  follows  that  x1*1   •  •  •   x1**  is  a  sum  of  cosines  of  even  multiples 

Ai  \ 

of  VAT,  the  highest  being  given  by  (62).     From  properties  1,  4,  and  (a)  it 

follows  that  2/1'  •  •  •  t/""'  is  a  sum  of  cosines  of  even  multiples  of  VAr, 

\  V 

the  highest  multiple  being  given  by  (63).  From  properties  2,  5,  and  (b)  it 
follows  that  z*'  •  •  •  z1*""  is  also  a  sum  of  cosines  of  even  multiples  of 

*i  V' 

VAr,  the  highest  multiple  being  given  by  (64).  Therefore  Pn  is  a  sum 
of  cosines  of  even  multiples  of  VAr,  and  from  (62),  (63),  (64),  and  (f) 
it  follows  that  the  highest  multiple  is  n+1. 

Similarly,  it  can  be  shown  that  Qn  is  a  sum  of  sines  of  even  multiples  of 
VAT,  the  highest  multiple  being  w+1. 

In  this  case  J?n  =  0.     Therefore,  for  n  odd,  equations  (59)  become 


d?yn  |  2dx"     (i 
dr2          dr 


(69) 

UT 

d2zn 


dr2 


+Azn=  -2VAcdnsinVAT. 


OSCILLATING    S  VI  KI.I.l  I  KS  —FIRST   METHOD. 


173 


The  solution  of  these  equations  which  satisfies  the  periodicity  condition  and 
the  conditions  zn  =  z'n  =  Q  at  T=0,  is 


I  i 

£»o 


„ 
" 


/ 


2  (8j  +  2jt-UAt-(8jt-UA  +  1 


»<.._  ±4; 


" 

/  . 
V 


n+l\ 
TV1 

n+l\ 
"TV* 


7(h 


Since  .A>1  these  denominators  can  not  vanish  for  any  integral  j. 

It  is  obvious  that  in  practice  it  is  not  necessary  to  refer  to  the  differ- 
ential equations  at  each  step.  The  most  convenient  method  to  follow  is 
to  substitute  as  many  terms  of  (54)  in  the  right  members  of  (12)  as  will  be 
required  in  carrying  the  computation  to  the  desired  order  in  e,  and  to  arrange 
the  results  as  power  series  in  «  of  the  form 


and  similar  series  for  the  other  equations.  From  the  P»,  Qn,  and  /?„  the 
A(*\  B™,  and  O™  can  be  computed  sequentially  with  respect  to  n  without 
explicit  reference  to  the  left  members  of  the  differential  equations.  The 
coefficients  of  the  solutions  are  given  by  (68)  and  (70).  The  whole  process 
is  unique  and  can  be  continued  as  far  as  may  be  desired. 

88.  Additional  Properties  of  Orbits  of  Class  A.  —  It  will  be  observed 
that,  so  far  as  the  computations  have  been  carried,  x,,  y,,  and  z,  carry  c'+l 
as  a  factor  and  that  8,  carries  c'  as  a  factor.  We  shall  prove  that  this  is  a 
general  property. 

Suppose  it  is  true  for  j  =  0,  .  .  .  ,  n  —  l,  and  consider  the  question 
for.;  =  /i.  The  terms  of  order  n  are  defined  by  equations  (59).  In  P. 
there  are  terms  y',8m.j-  It  follows  from  the  assumed  properties  of  the  y, 
and  S,  that  this  term  carries  c*+'  as  a  factor.  Similarly  the  x',  $„_,  occurring 
in  Q,  carry  c^1"1  as  a  factor.  Now  consider  the  general  term  (60).  It 
follows  from  the  assumed  properties  of  x,,  y,,  z,,  and  S,  that  this  term 
carries  c  as  a  factor  to  the  power 


It  follows  from  (f)  that  N  =  n+l,  and  therefore  this  property  is  general. 


174  PERIODIC   ORBITS. 

In  order  to  obtain  the  coordinates  in  the  physical  problem,  we  must 
replace  e  by  e  and  multiply  x,  y,  and  z  by  e'  [equations  (10)]  .  Then  e'  and  c 
occur  in  every  term  in  x',  y',  and  z'  to  the  same  degree,  and  are  equivalent 
to  a  single  parameter.  That  is,  without  loss  of  generality  we  may  put  c  equal 
to  unity,  and  the  value  of  t  will  determine  the  dimensions  of  the  orbit.  Or, 
if  we  put  e'  equal  to  unity,  c  will  determine  the  dimensions  of  the  orbit.  It 
follows  from  these  results  that  the  coordinates  and  8  are  expansible  as  power 
series  in  c,  and  the  solutions  could  have  been  derived  in  this  way  without  the 
introduction  of  e  and  e',  but  the  discussion  would  have  been  less  simple. 

The  explicit  expressions  for  the  periodic  solution,  so  far  as  they  have 
been  worked  out  in  (29),  (56),  and  (58),  are* 


sinVJr]e' 


_n  9     r    3£2(1-3A  +  14A2)        -I  ,2  , 

~  16  A2  L(1+2A)(1-7A  +  18A2)  " 


(71) 


Since  x'  and  y'  are  sums  of  cosines  and  sines  respectively  of  even 
multiples  of  VAr,  and  since  z'  is  a  sum  of  sines  of  odd  multiples  of  VZr,  it 
follows  that  x'  and  y'  are  periodic  with  half  the  period  of  z'. 

Since  the  relations 

X'(r)  =  x'(-r),  y'(r)  =  -y'(-r),  z'(r]  =  -z'(-r) 

are  satisfied,  the  orbits  are  symmetrical  with  respect  to  the  z-axis,  as  was 
shown  in  the  existence  proof. 

Let  T  equal  half  the  period.     Then 

x'(r)  =  o/(T+r),  y'(r}  =  2/'(T  +  r),  z'(r}  =  -z'(T  +  r). 

Therefore  the  orbits  are  symmetrical  with  respect  to  the  x'y'-pl&ne.     Simi- 
larly, since 

z'(r)=z'(T-r),  y'(r)=-y'(T-T),  z'(r]  =z'(T-r), 

the  orbits  are  symmetrical  with  respect  to  the  x'z'-plane. 

It  follows  from  the  form  of  (71)  that  a  change  of  the  sign  of  e'  is 
equivalent  to  changing  T  by  a  half  period. 

The  period  of  the  solutions  in  r  is  2w/\rA,  but  it  follows  from  the 
last  equation  of  (10)  that  in  the  time  t  the  period  is 


p_    T  2a-  fi         9     f    3  £2  (1-3  A  +  14  A2)          ~-| 

"  V4l6~^  1(1+2^X1-7  A  +  18  A2) 

*The  x',  y',  and  z'  are  the  actual  coordinates  and  not  their  derivatives. 


OSCILLATING    SATKM.H  KS       KIHST    \IKIIHU).  175 

It  is  found  from  (71)  that  the  equation  of  the  projection  of  the  orbit 

on  the  j-y-pkuie  is.  up  to  terms  of  the  fourth  «l«^rcr  in  «', 


z'4  «        T       ,   „_          9BV'  -„> 

h4A(l+2A)J  "(1-7A  +  18A')'' 

_  OD-'J 

This  is  the  equation  of  an  ellipse  whose  center  is  at  x'=          ^~\       ,  y'  =  0, 
and  whose  semi-axes  are 

3fl(l+3A)e"  :Uhn  ,-A. 

4  A  (1-7  A  +  18  A')'          vCT(i-7A  +  18A!)' 

The  equation  of  the  projection  of  the  orbits  on  the  yV-plane    is 
approximately 


-&Bz'  A:'  /7KN 

-  (1-7A+17AV  V1     -?- 

This  is  tin-  equation  of  a  figure-of-eight  curve  with  its  center  at  the  origin, 
touching  the  i/'-axis  at  no  other  point,  and  having  two  other  intersections 
with  the  z'-axis. 

The   equation  of   the  projection  of  the  orbit  on    the   z'z'-plane   is 
approximately 

-1  (1+3,4) 


The  orbit  is  a  parabola  whose  axis  is  the  z'-axis,  and  whose  vertex  is  at 

X'=  _  93(1  -A)e"  >=f  =  Q 

(l+2Ani-7A  +  18A>) 

Only  that  part  of  the  parabola  for  which  z'*<t'*/A  belongs  to  the  orbit, 
the  infinite  branches  having  been  introduced  in  eliminating  r. 

It  is  at  the  vertex  of  this  parabola  that  the  orbit  has  the  double  inter- 
section with  the  z'-axis.  In  all  cases  1  —  A  <  0,  and  therefore  1  —  7  A  +  ISA1  >  0. 
It  is  seen  from  (35)  that  B  is  positive  for  the  point  (a).  Hence  these  orbits 
intersect  the  z'-axis  between  (a)  and  the  finite  mass  ». 

It  follows  from  (35)  and  the  second  equation  of  (4)  that  B  is  negative 
for  the  point  (6),  at  least  if  n  is  small.  Hence  these  orbits  intersect  the 
z'-axis  between  the  point  (6)  and  the  finite  mass  M-  Similarly,  those  orbits 
near  (c)  intersect  the  z'-axis  between  (c)  and  the  finite  body  1  —  n. 

The  vertices  of  these  parabolas  are  the  ends  of  the  ellipses  whose  axes 
are  given  in  (74).  It  is  seen  from  the  expressions  for  the  coordinates  of  the 
centers  of  the  ellipses  and  the  ends  of  the  parabolas,  that  the  distance 
from  the  vertices  of  the  parabolas  to  the  centers  of  the  ellipses  is 


44(1  -74  +  ISA')  ' 

It  follows  from  the  signs  of  B  that  the  orbits  in  all  cases  open  out  away  from 
the  points  of  equilibrium  near  which  they  lie. 


176 


PERIODIC    ORBITS. 


From  the  properties  which  have  been  derived  it  is  possible  to  infer  the 
geometric  character  of  these  orbits.  In  a  general  way  they  have  the  shape 
of  the  handles  of  ice-tongs,  one  of  the  two  handles  being  situated  on  one  side 
of  the  £r;-plane,  and  the  other  symmetrically  on  the  other  side  of  this  plane. 
The  place  of  the  hinge  is  where  they  cross  the  £-axis.  In  the  case  of  the 
points  (a)  and  (6)  they  open  toward  the  finite  mass  n,  and  in  the  case  of 
(c),  toward  the  finite  mass  1—  /j.. 

In  Fig.  3  an  orbit  of  each  class  is  shown.  No  attempt  has  been  made 
to  represent  them  to  scale  for  any  particular  case,  but  the  figures  show  their 
general  positions  and  the  directions  of  motion  in  them.  The  curves  are 
drawn  on  elliptic  cylinders  to  make  the  figures  as  clear  as  possible. 


(C) 


0 


(I) 


fj- 


FIG.  3. 

89.  Application  of  Jacobi's  Integral  to  the  Orbits  of  Class  A. — The 

original  differential  equations  admit  the  integral  (13),  which  holds  for  all 
orbits,  and  therefore  in  particular  for  the  periodic  orbits  which  we  are  dis- 
cussing. It  has  already  been  seen  that  it  plays  an  important  role  in  the 
proof  of  the  existence  of  the  periodic  solutions  when  we  start  from  general 
initial  conditions,  and  we  shall  now  show  that  it  is  almost  equally  important 
in  the  construction  of  these  solutions.  We  are  illustrating,  in  a  particularly 
simple  problem,  a  new  and  valuable  use  to  which  integrals  may  be  put. 
The  explicit  form  of  the  integral  (13)  is 


+J5(-2z2+3?/2+322)ze+ 


.1  = 


constant. 


(77) 


OSCILLATING    SATELLITES —FIRST   METHOD.  177 

If  we  substitute  the  solutions  i49),or  rather  the  equivalent  series  for  .r.  //,  z, 
and  their  derivatives,  and  the  >cric-  for  5  in  (77),  and  then  arrange  as  a 
l>ower  series  in  c.  we  have 

— 'W 
dr/f 

«"+  •  •  •  =  constant. 


+Fn( 


,7s, 


Since  this  is  an  identity  in  e,  each  coefficient  is  a  constant.  We  shall  now 
work  out  the  form  of  the  general  term  of  (78).  It  is  seen  from  (77)  that 
the  function  F,  will  contain  terms  of  the  types 

$!*£=/,  i'^N,  '^-'         and  (60). 

dr    dr  dr    dr  dr    dr 

It  follows  from  the  properties  of  x,  y,  and  z  that  the  terms  of  the  first  three 
types  are  zero  unless  n  is  even. 

Now  consider  the  term  of  ty{>e  (60).  All  the  properties  of  its  exponents 
and  subscripts  are  the  same  as  when  it  belonged  to  Pn  except  that  in  the 
relation  (f)  the  —  1  in  the  left  member  must  be  replaced  by  —  2.  Hence 
we  see  that  this  term  is  also  identically  zero  unless  n  is  even. 

Since  /•'„  involves  only  even  powers  of  the  y,  and  z,,  it  is  a  sum  of  cosines 
of  even  multiples  of  V^Ar.  It  follows  from  the  relations  (a),  .  .  .  ,  (f) 
(the  last  one  modified  as  indicated)  that  the  highest  multiple  is  n+2.  Hence 
we  may  write 


+  •  •  •  +D£),cos(n+  2)  vTr  =  constant. 
Since  Fn  is  identically  a  constant,  we  have 

D;«-  constant,     D'™=  •  •  •   =D'£\=0.  (80) 

The  quantities  D't{m},  .  .  .  ,  D'$t  depend  upon  a<;>,  .  .  .  ,  a';';  0<",  .  .  .  ,  ft"; 
7;°  ,  .  .  .  ,  7".  Suppose  all  the  a,,  0,,  7,  up  to  <-»,  ft-*,  y»-'>  have 
been  computed  and  are  known  to  be  accurate.  Equations  (80)  can  then 
be  used  in  two  ways,  as  we  shall  show.  First,  they  test  the  accuracy  of 
the  computation  of  the  a?1"",  ff~l\  y("~l\  for  these  quantities  must  have 
such  values  that  the  equations  shall  be  satisfied.  And  secondly,  we  can 
compute  the  a7~",  P*~"  from  equations  of  the  type  of  (69),  and  then  find 
the  75°  and  6.  directly  from  (80)  without  referring  to  the  differential  equations 
of  the  type  of  (67). 

The  first  use  is  obvious  and  we  need  to  consider  further  only  the  second. 
We  are  working  under  the  hypothesis  that  n  is  even.  Therefore  the  a*' 
and  the  0*'  are  identically  zero.  It  is  seen  from  (77)  that  the  only  terms 
which  can  introduce  the  7^'  and  6.  are 

(81) 


178  PERIODIC    ORBITS. 

From  the  form  of  zn  given  in  (68),  we  have 


cos(n+2)  VTr 


j, 


(82) 


Therefore  equations  (80)  become 


D;W  =  2  c  vT  7<B)+c2  8.+J5J0  =  constant, 
D'™  =  4  c  <SA  73"'  -  c2  5B+Z><n)  =  0, 


(83) 


where  D^J  •  •  •  i  ^"+2  are  known  constants  depending  upon  a^",  .  .  .  ,  of~"; 

0(1)  oW-1).        (0)  (n-2) 

Pj   t  •  •  •  ;  Pj        >     I)   }   •  •   '  )    ij 

The  last  n/2  equations,  beginning  with  the  last  one,  can  be  solved  for 
7^+!  ,  .  .  .  ,  7gB)  in  order.  Then  the  second  equation  gives  dn  uniquely.  The 
results  of  these  solutions  are 


<n> 


W      =  (n) 

- 


: 

D? 


All  the  constants  are  uniquely  determined  except  7in),  which  is  defined 
by  the  condition  that  z'n  shall  be  zero  at  r  =  0.    This  condition  gives 


(85) 


Thus  we  see  that  in  orbits  of  Class  A  we  can  suppress  the  z-equation, 
if  we  wish,  and  compute  the  7^  from  the  integral;  or,  we  may  use  the 
integral,  step  by  step,  as  a  check  on  the  computations. 


OSCILLATING   SATELLITES— FIHST   METHOD. 


179 


90.  Numerical  Examples  of  Orbits  of  Class  A.  No  periodic  orbits  of 
this  class  have  -<>  far  been  published.  It  is  clear  that  it  is  practically 
impossible  to  discover  them  by  numerical  experiment.  We  shall  suppose 
the  ratio  of  the  finite  ma.--e-  i-  ten  to  one,  or  1  —  A»  =  10/1 1,  n  =  1/11.  Then, 
in  the  computation  of  the  coefficients  of  the  series  for  the  .solutions  in  the 
vicinity  of  the  points  (a),  (6),  and  (c),  the  following  results  are  found: 


Coefficient 

Point  (a).                           I'oint  (6). 

Point  (c). 

rj"   (EquatioiiH  (4)] 

+  0.347                         +  0.282 

+  1.947 

r,'"   (Equations  (4)] 

+  1.347 

+  0.718 

+0.947 

A    [Equation  (15)] 

+  2.548 

+  6.510 

+  1.082 

a*    [Equation  (19)] 

+  2.811 

+  6.820 

+  1.144 

P'    [Equation  (19)] 

+  3.359 

+  11.330                         +0.226 

n    [Equation  (28)] 

+  2.657 

+  3.990                         +2.014 

HI    [Equation  (28)] 

-  0.747 

-  0.397                         -3.091 

K    (Equation  (35)] 

+  6.548 

-10.961                         -1.136 

C    [Equation  (35)] 

+  18.283                         +55.740                         +1.196 

-'•US                t(lM\\ 

-  0.316 

+  0.090                         +0.249 

4A(1+2A)    ' 

+3BU+3A) 

+  0.151 

-  0.036                         -0.230 

A  A  1  1       7  A    t   1  Q  A  S  \     •  \*^*/  J 

*•  /i  ^  i  ^  /  **t  -|~  »o  **  / 

-3B 

-  0.112 

+  0.018                         +0.226 

V/A(1—  7A+18A*) 

r3B»(l+3A)        -I 

-  0.037 

-  0.020 

-0.002 

64A'/'Ll-7A+18A«        _P 

-9  T    3B«(1-3A  +  14A«)          ^ 

+  0.184 

+  0.467 

+0.001 

16.*lL(I"i"*«)  (1  —  t  A  ~\~\oA)           _} 

^-period 

3.936(1+0.184«'«+--) 

2.463(1+0.467  «••+••) 

6.041(1+0.001  «"+••) 

From  these  results  we  find  that  the  solutions  in  the  neighborhood  of  the 
three  points  of  equilibrium  are 

x'  =  [-0.316+0.151  cos  2v/Ar]«"+  •  •  •  , 


(a) 


«"+ 


7/'  =  [-0.112sin2v/Jr]€'1+ 
2' =  [+0.626  shWArlt'+f- 0.037  (3sinvGfr-i 
x'  =  [+0.090- 0.036  cos2 VA r]  «"+   •  •  •  , 
'  =  [+0.018  sin2v/AT]  e"+ 
z'  =  [ +0.392  sin  VAr] «'+ [  -  0.020  (3  sin  VAr  -  sin  3  vdr)] «"+ 

x'  =  [+0.249- 0.230  cos2VJT]«'*+  •  •  •  , 


(6)  \y'  =  [+0.018  sin2v/ZT]e"+  •  •  •  ,  (86) 


(c)  , 

'  =  [+0.961sinVXr]«'+[-0.002(3sinVAT-8in3>/XT)]«"+ 

If  we  regard  the  motion  of  the  finite  bodies  as  direct,  and  consider  the 
projections  of  the  motion  of  the  infinitesimal  body  upon  the  zV-plane,  we 
find  that  in  all  cases  the  motion  in  orbits  of  Class  A  is  retrograde. 


180 


PERIODIC    ORBITS. 


91.  Construction  of  a  Prescribed  Orbit  of  Class  A.  —  Suppose  the  masses 
of  the  finite  bodies  are  given.  Then  a  periodic  orbit  of  Class  A  for  the 
infinitesimal  body  may  be  defined  (1)  by  the  place  it  crosses  the  a/-axis, 
(2)  by  the  y'  or  z'-component  of  velocity  with  which  it  crosses,  (3)  by  the 
greatest  value  of  the  y'  or  z'-coordinate,  (4)  by  the  constant  of  the  Jacobian 
integral,  and  (5)  by  the  period.  It  is  understood,  of  course,  that  these 
various  quantities  are  arbitrary  only  within  such  limits  that  the  series  for 
the  coordinates  converge. 

For  «'  =  0  the  Jacobian  constant  C  and  the  period  have  definite  values 
depending  only  upon  /JL.  The  increments  to  these  values  and  all  the  other 
defining  quantities  enumerated  above  can  be  developed  as  power  series  in 
t,  vanishing  with  e'.  These  series  are  odd  or  even,  depending  upon  which 
other  quantity  is  taken  as  defining  the  orbit.  If  s  represents  any  of  these 
quantities,  we  can  write 

s  =  S1e'+S2e'2+s3e'3+ 

where  the  coefficients  sl  ,  s2  ,  s3  ,  .  .  .  are  constants  which  depend  upon  /z  alone. 
If  s  is  assigned  numerically  the  inversion  of  this  series  gives  e.  This  value 
of  e'  substituted  in  (71)  gives  the  desired  orbit.  Thus,  the  methods  which 
have  been  developed  not  only  prove  the  existence  of  the  periodic  orbits 
and  give  convenient  processes  for  constructing  them  and  testing  the  accu- 
racy of  the  computations,  but  they  furnish  a  ready  means  of  finding  any 
particular  orbit  that  may  be  desired. 


92.  Existence  of  Orbits  of  Class  B.  —  For  e  =  0  the  coordinates  in  these 
orbits  are  given  by  (30).  Therefore  at  =  a2  7^  0,  a3  =  a4  =  c=  0  in  (38). 
Sufficient  conditions  that  the  solutions  (39)  shall  be  periodic  with  the  period 
2ir/<r  are 


^)  -  w,(0)=0 

f)  -  *(0)-o, 


(i=l,  ...  ,4), 
z'&L}-  z'(0)=0. 


(87) 


If  we  let 


z  = 


the  integral  (13)  may  be  written 


,   5,  e]  -F^ 


o,,  a4,  0,  y,  8,  e]=0. 


(88) 


OSriU.ATIN.,    SATKI.UTKS       I  IRST    METHOD.  181 

This  equation  i^  >ati>!inl.  at  r  =  2*/ff,  by  r,=  f  =  f'=0;  and  we  find  from 
the  explicit  form  of  /•'  and  the  traii>formation  (33)  that,  for  these  values  of 

the  variables  and  «,•  •  •  •  =  a1  =  s  =  «  =  o, 


Hut  from  (-(iimtions  (19)  and  (28)  we  have 

S+1+2A          2<r 
2<r 


•Therefore  n'o*-l-2A-ol+nt(A-l),  which  is  always  positive.  Conse- 
quently, for  r  =  2ir/a,  (88)  can  be  solved  uniquely  for  r,  in  terms  of  a,,  y, 
5,  t,  r,  ,  P,,  t>4,  f,  f',  and  this  solution  vanishes  for  r,  =  vt  =  vt  =  f  =  f  '  =  0. 
Hence,  if  we  impose  the  condition  that  r,  =  r,  =  vt  =  f  =  f  =  0  at  T  =  2rA, 
the  equation  r,  =  0  at  r  =  2r/a  will  be  satisfied.  Therefore  the  first  equation 
is  redundant,  and  it  will  be  suppressed. 

It  will  be  shown  that  the  orbits  of  Cla.vs  B  liejn  the  ary-plane.  It 
follows  from  the  form  of  the  last  equation  of  (34)  and  the  initial  values  of  z 
and  z'  that  I\  and  Pt  contain  7  as  a  factor.  Therefore  the  last  two  equations 
of  (88)  contain  7  as  a  factor.  The  explicit  form  of  the  next  to  the  last  one  is 

'  ~P'(0)  =  VA  8in 

When  VA/tr  is  not  an  integer  the  only  solution  of  this  equation,  vanishing 
with  the  parameters  in  terms  of  which  the  solution  is  made,  is  7  =  0.  The 
case  where  vCT/a  is  commensurable  will  be  considered  in  connection  with 
the  orbits  of  Class  C.  Therefore  z  =  0,  and  the  orbits  are  plane  curves. 

Necessary  and  sufficient  conditions  for  the  existence  of  the  ixriodic 
solutions  of  Class  B  reduce  to  (87),  where  t  =  2,  3,  4.  The  explicit  forms 
of  these  equations  are 


(90) 


We  have  three  ecjuations  to  satisfy  and  five  arbitrary  parameters, 
besides  «,  at  our  disposal.  The  parameter  o,  enters  only  in  the  combination 
a,+a,,  and  since  a,  is  as  yet  subject  only  to  the  condition  that  it  shall  not 
vanish,  we  may  let  it  absorb  o, .  \Ve  may  determine  /„ ,  which  enters  in  the 


182  PERIODIC    ORBITS. 

definition  of  T,  so  that  at  r  =  0  we  shall  have  x'  =  0,  a  condition  which  is 
fulfilled  in  all  closed  orbits  in  which  the  coordinates  have  continuous 
derivatives.  By  (33)  this  condition  becomes 


—  ffV—  1  o,+p(a3  —  <0=0, 

which  we  may  regard  as  eliminating  a,  . 

We  now  consider  the  solution  of  (90)  for  a,  ,  a4  ,  and  6  as  power  series 
in  e,  vanishing  with  «.  The  determinant  of  the  linear  terms  in  aa,  a4, 
and  6  is 

(91) 


which  is  not  zero.  Therefore  equations  (90)  have  a  unique  solution  for 
a3,  a4,  and  5  as  power  series  in  «,  vanishing  with  «.  When  these  results 
are  substituted  in  the  first  four  equations  of  (39),  the  latter  become  power 
series  in  «  which  are  periodic  in  r  with  the  period  2ir/(r. 

It  will  now  be  shown  that  all  orbits  of  this  class  are  symmetrical  with 
respect  to  the  or-axis.  We  choose  ta  so  that  we  have  y  =  x'  =  0  at  r  =  0. 
Therefore  it  follows  from  equations  (33)  and  from  the  initial  values  of  the 
w,  that  otj  =  0,  a3  =  a4  .  Necessary  and  sufficient  conditions  that  these 
S3'mmetrical  solutions  shall  be  periodic  are 


It  follows  from  (33)  that  these  equations  are  equivalent  to  ul  =  ut,  ut  =  u4 
at  T  =  ir/ff.    The  explicit  expressions  for  the  latter  become 


(92) 


/  t      \ 

,  (a,,  «,  •), 

-i   ,     n/  ,  N 

J +€  Q3(«»,  3,  e). 


The  determinant   of   the   coefficients   of  the   terms   which   are  linear  in 
a3  and  5  is 

,  -  r   Z»—          -2r—  ~\ 

1v^T  [e    °  —e     "J> 


which  is  not  zero.  Therefore  equations  (92)  can  be  solved  uniquely  for  a, 
and  d  as  power  series  in  t,  vanishing  with  e.  Since,  for  a  given  value  of  e, 
there  is  but  one  unrestricted  orbit  of  this  class,  and  since  there  is  also  one 
which  is  symmetrical  with  respect  to  the  x-axis,  it  follows  that  all  orbits 
of  this  class  are  symmetrical  with  respect  to  the  z-axis. 

The  orbits  of  this  class  all  re-enter  after  one  revolution,  for  if  we  impose 
the  conditions  that  they  re-enter  after  v  revolutions,  we  find  the  solution  is 
unique.  Since  it  includes  those  which  re-enter  after  one  revolution,  it 
follows  that  all  orbits  of  this  class  re-enter  after  precisely  one  revolution. 


OSCILLATING    SATELLITES — FIRST   METHOD. 


ISM 


93.  Direct  Construction  of  the  Solutions  for  Class  B.     It   has  been 

shown  that  in  the  periodic  orliits  of  ( 'la>s  H  the  coordinates  are  uniquely 
developable  in  M-rir-  of  the  form 


w,=  2  uJV 


i-o 


l,  -..  ,4), 


5=2  «,«', 


(93) 


where  the  u\"  are  periodic  functions  of  r  with  the  period  2»/<r,  and  where 
the  d,  are  constants. 

\\  »•  have  seen  that  without  loss  of  generality  we  can  put,  at  r  =  0, 


Wl  =  ««  =  «l=2»  "ta- 

ll follows  from  (33)  and  (93)  that  we  have  also 


,?.*••'• 


y=  2  K, «'. 

1-0 


1 94 ) 


(95) 


Upon  substituting  (93)  in  (34),  arranging  as  power  series  in  «,  and  equating 
coefficients  of  corresponding  powers  of  «,  we  obtain  a  series  of  sets  of 
differential  equations  from  which  the  u\n  can  be  determined. 
The  terms  independent  of  c  are  defined  by 


The  solution  of  these  equations  which  satisfies  the  periodicity  conditions 
and  the  initial  conditions  is 


-»V^I 


From  these  results  and  (33)  we  get 

x,  =  a  cos  ITT,  y0  =  —  na  sin  or. 

The  terms  of  the  first  degree  in  t  are  defined  by 


(96) 


(97) 


-        n^-iiu" 


-_ 

^l       2(mp+n<r) 


2(mp+rur) 


dr 


. 


4(m<r— np)  2(mp+n<7) 


-! 


4(m<7-np) 


(98) 


184  PERIODIC    ORBITS. 

The  conditions  that  the  solutions  of  these  equations  shall  be  periodic  with 
the  period  2ir  /a  are  that  the  right  members  shall  be  periodic  with  this 
period,  that  the  right  member  of  the  first  equation  shall  not  contain  the 
term  effV=rTT,  and  that  the  right  member  of  the  second  equation  shall  not 
contain  the  term  e~"v^:jT.  The  first  condition  is  satisfied,  and  on  referring 
to  (97)  we  see  that  the  second  and  third  conditions  can  be  satisfied  only 
by  5,  =  0.  Therefore  we  put  5:  equal  to  zero. 

Upon  substituting  from  (97),  equations  (98)  become  in  full 


du™  _  <,.,/— r  um  _  i   3m.Ba*[(n'-2) - (n2+2)cos2gr]   ,    3n^a2sin2o-T  ( 
dr  8(ma-  —  np)V—l  4(mp+w<r) 


,  O.v/—  j-  M«)  =  _  3mJBa2[(n.2-2)-(n2+2)cos2q-T]  _, 


dr 

(99) 


dr 


_   U( 


_  _  3nJga2[(n2-2)-(n2+2)cos2q-T]       3nga2sin2g-r 


du(?  .    Ma>  =  _,    3  nBa?  [  (n2  -  2)  -  (n2  +  2)  cos  2  ar\  __  3n.Ba2sin2<rT 
dr  8(m<r  —  np)  ±( 


The  solution  of  the  first  equation  of  (99)  has  the  form 

-V^Ib™  sin  2<rr,  (100) 


where  cj"  is  the  arbitrary  constant  of  integration.  Upon  substituting  this 
expression  in  the  first  of  (99)  and  equating  coefficients  of  like  functions 
of  T,  we  get 


(101) 


o>  =  i 

8(m«r-np)(r  ' 


a«,, 


6n,  =  _  , 

— 


2(mp+na)<r 


4(m<r  —  np)a        4 


It  follows  from  the  form  of  (99)  that  the  solution  of  the  second  equation 
can  be  obtained  from  that  of  the  first  by  changing  the  sign  of  V  —  i. 
Therefore 


0(i>_0a> 

U10          "20  ) 


The  solution  of  the  third  equation  of  (99)  has  the  form 

cos  2  ffr  +  &S'  sin  2  or, 


OSCILLATING    SATELLITES — FIRST   METHOD.  185 

where,  because  of  the  periodicity  condition.  ri"  =  0.     \\  e  find  by  substitution 
in  the  differential  equations  that 


=  +    _3nfio1(nt-2) 

a«  = 


8(m<r— »p)p 

;*(n'+2)p  3nflaV 


2(rop+n<r)(4<r*+pt) 

w=  -/)"'=  -  ZnBcfp 

'  " 


4(wur-np)(4<r'+p*)      4(mp  +  n<r)(4<r*+p') 


(103) 


Since  x  =  a  and  j/  =  0  at  r  =  0  for  all  values  of  t,  we  have  x,(0)  =  y,(0)  =  0. 
Upon  substituting  uju,  .  .  .  ,  u\"  in  (33)  and  applying  these  conditions,  we  find 


].  (104) 

Then  the  expressions  for  x,  and  j/,  become,  by  (33), 


(105) 


sin2<rr, 


In  order  to  see  how  the  construction  goes  in  general,  the  process  must 
be  continued  one  step  further.  The  differential  equations  which  define  the 
next  terms  are 


—  0  v  —  1U.     —  ~T~  (TV  —  1  U,    0,~\ r — /I —  ^  — ;T7 — 

dr  2(mff  —  np)V^  2(mp+n<r) 

+ 

+<rv^Tttia)  =  -  (rv^^Tui0'^- 


-np)  V- 
o> 


2(wur-np)V-l          2(mp+n<r) 


t       , 
(nur—  np)  V—  1          4  ( 


_        =  _          -  2ayr, 


-2'niff-np)  2(mp+n<r 


- 
dr     "     4  "  2(ro<r-np) 


mc—np  4(mp+n<r) 

-'•  -i)°' 


,    n(7[2x;- 


m<r—  np 


lor,, 


186  PERIODIC    ORBITS. 

In  order  that  the  solutions  of  these  equations  shall  be  periodic  with  the 
period  2,-ir/a,  the  right  members  must  be  periodic  with  this  period,  the  right 
member  of  the  first  equation  must  not  contain  the  term  effV^r,  and  the 
right  member  of  the  second  equation  must  not  contain  the  term  e~"^f^lT, 
The  first  condition  is  satisfied  as  the  equations  stand.  The  expression 
eov=rr  enters  through  w<0),  x0x,,  y^y^  x^y^  and  x,y0.  The  sum  of  its 
coefficients  must  be  put  equal  to  zero;  this  condition  determines  52  by  the 
equation 

3  mB  [4(a{?  +  off)  +2«  +Q  +n 


(107) 


5  =  _. 

2(ma—np)p 

0 


a{10+ 


2  (rap + na)  a 

3raa2C(2-n2)  _  3na2C(4-3n2) . 


8  (ma —np)<r        32  (rap + no-) 

This  disposes  of  all  the  arbitraries  appearing  in  the  equations,  and  the 
third  condition  remains  to  be  satisfied.  Upon  comparing  the  first  and  second 
equations  of  (106),  we  see  that  the  signs  of  the  [  ]  in  the  second  are  opposite 
the  signs  of  the  corresponding  terms  in  the  first,  and  that  corresponding 
\  j  have  the  same  sign  in  the  two  equations.  It  is  observed  that  the  [  ] 
are  sums  of  cosines  of  multiples  of  or,  while  the  -J  [  are  sums  of  sines 
of  the  same  arguments.  Since  62  enters  in  the  second  equation  with  the 
sign  opposite  to  that  in  the  first,  it  follows,  as  a  consequence  of  the  prop- 
erties of  [  ]  and  ]  \ ,  that  the  condition  on  the  second  equation  is  satisfied 
by  the  same  value  of  52  as  that  which  satisfies  the  condition  on  the  first. 

We  now  proceed  to  the  general  term.  Suppose  xl  ,  .  .  .  ,  xv^; 
yl ,  .  .  .  ,  yv-j.  have  been  computed,  and  that  they  have  been  found  to 
have  the  following  properties : 

1.  The  Xj  are  sums  of  cosines  of  multiples  of  or. 

2.  The  yt  are  sums  of  sines  of  multiples  of  ar. 

3.  The  highest  multiple  of  ar  in  x,  and  y,  isj+1. 

4.  The  [  ]  is  an  even  function  of  y. 

5.  The  \   \  is  an  odd  function  of  y. 

The  equations  defining  the  coefficients  of  e"  are 


2(rap+Wtr) 
> 

(108) 


_. 

dr  2  (ma  —  np)      2  (rap + no) 


2  (ma  —  np)      2  (rap  +  no) 
where  the  &, ,  .  .  .  ,  dy^  are  included  in  [  ]  and  {    } . 


OSCILLATING    SATELLITES-  FIRST   METHOD. 


187 


It  follows  from  properties  1,  2,  4,  and  5  that  [  ]  is  a  sum  of  cosines  of 
multiples  of  ar,  and  that  :  I  is  a  sum  of  sines  of  multiples  of  or.  The 
general  term  of  [  ]  is 

r'  =  xx'  •••<' •!£'•••  I/?'*;-  <109) 

I  C  I  .' 

The  exponents  and  subscripts  of  thi-  expression  satisfy  the  conditions: 

(a)  n[+  •  •  •  -f/v  is  an  even  integer  because  of  4. 

(b)  q  is  0  or  1 ,  since  &  enters  (34)  linearly, 
(c) 


The  product  x%  •  •  •  z£  is  a  sum  of  cosines  of  multiples  of  ar,  by  1. 

There  is  an  even  number  of  odd  /uj»  •  •  •  ,  AV,  by  (a).     Those  factors  $$  for 

j 
which  /ij  are  odd  are  sums  of  sines  of  multiples  of  or.    The  product  of  an 

even  number  of  such  factors  is  a  sum  of  cosines  of  ar.     It  follows  that 
T,  is  a  sum  of  cosines  of  multiples  of  or. 
The  highest  multiple  of  or  in  T,  is 


It  follows  from  (c)  that 


when    =  0. 


(110) 


Therefore  [  ]  is  a  sum  of  cosines  of  multiples  of  or,  the  highest  multiple 
being  v+1. 

It  can  be  shown  in  a  similar  way  that  •  |-  is  a  sum  of  sines  of  multiples 
of  or,  the  highest  multiple  being  v+1. 

Equations  (108)  can  be  written,  therefore,  in  the  form 


G+i  ~i     f'+' 

-  ,  J     (»)  •  I  .    -  /    , 

L  .4,  cosiffT  I— i  2  B. 
.«  J      L<-i 


M 
f_ 

dr 


aoV—l  .-»vr 

^ c 


[r+l 
S  ^{ 
1.0 

,.(»)  r'^1  ~\ 

u»   _  p  M  w  =  +  ;,  |  2  ^  ,{r)cos  tar 
ar  Li-o  J 

r'^1  n 

*  =  -  n  |  2  .4  i"cos  «Vr 
L«-o  J 


T        fr+' 

1-J21? 
J      1 1.1 


f+i  1 

2  ^'"sintffT  [•• 


(HI) 


where  the  A™  and  ^<"  arc  all  known  real  constants. 


188 


PERIODIC    ORBITS. 


In  order  that  the  solution  of  equations  (111)  shall  be  periodic,  we  must 
impose  the  conditions  that  the  coefficient  of  cav '  Tr  in  the  first  equation,  and 
of  e~"v=riT  in  the  second  equation,  shall  be  zero.  It  is  easily  seen  that  the 
two  conditions  are  identical,  and  they  uniquely  determine  5,,  by  the  equation 


- 
a 


(112) 


The  periodic  solutions  of  (111)  are  of  the  form 


= 


v+1  »+l 

•2   a™  COS  jar—  V—  1    S 


cos  jar  -V-l   2 


smjffT, 


r+l 


r+l 

«  +  2 


v+l 

S 


(113) 


where  cj"  and  c^'  are  arbitrary  constants  of  integration. 

Upon  substituting  (113)  in  (111)  and  equating  coefficients  of  corre- 
sponding functions  of  T,  we  find 


v 
1 


_mAT  +  j_B£_ 


JV+pP 
Then  equations  (33)  give 


(3  =  2, 


(3  =  1, 


j  =  l,  .  .  .  ,v+l). 


(114) 


v+l 


yv  =  nv/ -T  [clVv^'  -c^V^'] + 2  2i  [n  6J7 + m  Cl  «"««•• 


(115) 


OSCILLATING    SATELLITES — FIRST   METHOD.  189 

The  arbitraries  cj"  and  c™  arc  determined  l>y  the  conditions  that  x,  =  Q 
and  ?/r  =  Oat  r  =  0.     Upon  applying  these  conditions,  the  final  results  are 


-(«_,.(»)__   y    fn 
ci     —  cz    —         *•    la 
i-o 

x,  =  2[<+«S?l-2  Sl  [<+<>]  cosar+2 
j -o 

»  (  '  •>+» 

y,  =2n  2  [<+<?!  sin<n-+2  2  [n^+mig?]  sinjtrr. 


(116) 


The-e  expressions  have  the  properties  1  and  2.  Since  xt,  yt ,  xl ,  yl  also 
have  these  properties,  the  induction  is  complete  and  x  depends  only  upon 
cosines  of  multiples  of  <TT,  and  y  upon  sines  of  the  same  argument.  The 
orbits  are  therefore  symmetrical  with  respect  to  the  x-axis. 

94.  Additional  Properties  of  the  Orbits  of  Class  B. — It  is  observed 
that  xn  and  ya  are  homogeneous  of  the  first  degree  in  a,  and  that  xit  yt,  and 
6,  are  homogeneous  of  the  second  degree  in  a.  It  is  easily  proved  by 
induction,  making  use  of  the  general  term  (109),  that  xr  and  yf  are  homo- 
geneous of  degree  v+\  in  a,  and  that  5r+iis  homogeneous  of  degree  v+l  in  a. 
Consequently,  it  follows  from  (10)  that  the  actual  coordinates,  x'  and  y', 
carry  in  each  term  of  their  expansions  e'  and  a  to  the  same  degree.  That 
is,  «'  and  a  are  equivalent  to  a  single  parameter,  and  we  may  put  one  of 
them,  say  a,  equal  to  unity  without  any  loss  of  generality.  Then  the 
explicit  expressions  for  the  coordinates  become 

x'  =  [cosffT]e'+2[(aJ"+ ail,')  —  (0',"+ a» +a«  +  ««)  COSVT 

-f-Caij+aiy)  cos2<rr]  «'*+   •  •  •  , 

(117) 
y'  =  [-n  sin(7T]«'+2[+n(a'"  '  ~(I)  '  -(l)  '  -ah~- 


Since  n  is  a  positive  constant,  it  follows  that  in  all  cases  the  motion  in 
these  orbits  is  retrograde.  For  small  values  of  «'  the  orbits  are  approxi- 
mately elliptical  in  form,  the  axes  of  the  ellipses  coinciding  with  the 
x'  and  y'-axes. 

The  integral  can  be  applied  as  before  to  check  the  computation,  for 
it  has  the  form 


F(x,y,  ^>  ^ 


•  •  •   =  constant. 

Since  this  equation  is  an  identity  in  e,  each  F,  separately  is  a  constant.  It 
is  seen  that  F,  is  a  sum  of  cosines  of  multiples  of  <rr,  the  highest  multiple 
being  j  +  2.  Since  the  equation  is  an  identity  in  T,  the  coefficient  of  each 
cosjffT  (j  =  1  ,  .  .  .  ,  v  +  2)  separately  is  zero.  These  coefficients  involve 
the  a,"'  and  6,"'  linearly,  and  their  vanishing  constitutes  a  check  on  all 
the  oJi',  6,"'  from  the  beginning  of  the  computation  to  the  step  under 
consideration. 


190 


PERIODIC    ORBITS. 


95.  Numerical  Example  of  Orbits  of  Class  B. — In  Darwin's  memoir, 
cited  at  the  beginning  of  this  chapter,  there  are  a  few  examples  of  orbits  of 
this  class  in  the  vicinity  of  the  equilibrium  points  (a)  and  (6).  In  all  his 
computations  Darwin  took  one  finite  mass  ten  times  that  of  the  other. 
To  be  able  to  compare  the  results  of  this  analysis  with  his  orbits,  we  shall 
apply  the  formulas  for  the  same  ratio  of  the  masses.  This  was  the  ratio 
used  in  the  computation  of  §90,  and  the  first  part  of  the  table  given  there 
can  be  used  here. 

Upon  making  use  of  the  preceding  computations  and  (36),  (101),  (103), 
(104),  (107),  and  (117),  we  get  the  following  table  of  results: 


Coefficient. 

Point  (a). 

Point  (6). 

Point  (c). 

mp+nir       (36) 

+  3.085 

+  9.081 

+0.685 

ma—np        (36) 

-  6.121 

-14.467 

-4.263 

n'            (28) 

+  7.061 

+  15.920 

+4.058 

o£>          (101) 

+  0.905 

-  0.605 

-0.595 

mB(n'+2) 

+  0.540 

-  0.259 

-0.583 

Q       ,                                                                                                                                \\\}\.) 

nB 

+  1.682 

-  0.922 

-1.562 

2(mp+n<7)<7 

a',0          (101) 

-  1.142 

+  0.663 

+0.979 

by          (101) 

-  0.239 

+  0.057 

+0.385 

a«          (103) 

-  2.944 

+  4.691 

+0.872 

-3nfi(n'+2)p 

+  1.212 

-  1.772 

-0.121 

8  (ma  —  np)  (4<T1+p1) 

+3nB<r 

+  0.971 

-  0.489 

-1.116 

2(mp+n<r)  (4o-!+p:) 

<"          (103) 

+  2.183 

-  2.261 

-1.237 

b™          (103) 

-  1.687 

+  2.433 

+0.295 

c;0      (104) 

+  0.998 

-  2.488 

-0.019 

5,           (107) 

+  3.955 

+  8.553 

-1.407 

1 

We  find  from  this  table  that  when  the  ratio  of  the  finite  masses  is  ten  to  one, 
equations  (117)  for  the  three  equilibrium  points  (a),  (6),  and  (c)  are 


(a) 


(6) 


x'  =  [cos  <n-]e'+  [-4.078  +  1.996  cos  err  +2.082  cos  2<n-]e'2+ 
y'  =  [-2.657  sin  <rr]«'+  [-5.305  sin  err +  1.250  sin  2<n-]e/1+ 

x'  =  [cos  o-r]e'+  [+8.172-4.976  cos  or  -3.196  cos  2or]e'2+ 
y'  =  [-3.990  sin  (rr]e' +  [+19.855  sin  or- 1.478  sin  2<7r]e/2+ 

a;'  =  [coso-r]e'+  [+0.554- 0.038  cos  ar  -0.516  cos  2<rr]e'2+ 
^'  =  [-2.014  sin  <rr]e'+  [+0.077  sin  err  -0.276  sin  2(n-]e/2+ 


(118) 


OSCILLATING    SATELLITES  —  FIRST   METHOD. 


191 


96.  On  the  Existence  of  Orbits  of  Class  C.  -For  t  =  0  the  equations 
of  those  orbits  arc  ^iven  in  (32).  It  follows  from  these  equations  and 
(33)  that  in  this  case 


If  the  initial  conditions  are 


0,      z'  =  c+y, 
(t-1,  ....  4), 


(120) 


the  solutions  of  (34)  may  be  written  in  the  form 
w,  =  Pt(o,  ,  .  .  .  ,  a4  ,  y,  S,  <;  T) 
z  =  Pt(alt  .  .  .  ,a4,  y,  5,  t;  T),  (121) 

2'  =  P,(a,  ,  .  .  .  ,  a4,  y,  8,  t;  T), 

where  the  Pt  ,  P,  ,  and  P,  are  power  series  in  a,  ,  .  .  .  ,  a4,  X,  6,  and  «.    The 

conditions  for  iwriodic  solutions  with  the  period  —  £=  =  —  ^  are 

VA 


a 
l  5, 


,  e), 


l,  .  .  .  ,  a4  ,  y,  5, 
a1,  .  .  .  ,  a4  ,  7,  «, 


(122) 


Q,(o1,  .  .  .  ,  a4>  y,  5,  «), 
0=       +  (c+7)cos2*g(l+*)+«Q,(a1>  .  .  .  ,  a4  ,  7,  «,  e). 

Since  the  last  equation  of  (34)  carries  2  as  a  factor,  it  follows  that  the 
last  two  equations  of  (122)  are  divisible  by  c+y.  It  will  be  assumed  that 
this  factor  is  distinct  from  zero  and  is  divided  out.  We  shall  let  the  unde- 
termined constant  c  absorb  the  arbitrary  y.  Since  we  assume  that  c  is 
distinct  from  zero,  it  follows  from  the  integral  (13)  that  the  last  equation 
of  (122)  is  redundant  and  can  be  suppressed.  There  remain  five  equations 
whose  solutions  for  ^  ,  .  .  .  ,  a4  ,  and  5  as  power  series  in  «,  vanishing 
with  t,  will  now  be  considered. 

It  will  appear  in  the  course  of  the  work  that  we  shall  need  all  of  the 
terms  of  the  first  degree  in  «,  and  all  of  those  of  the  second  degree  which  arc 
not  periodic.  We  integrate  equations  (34)  as  power  series  in  t,  introducing 
5  in  the  combination  (l  +  S)r,  and  o,  ,  .  .  .  ,  a4  by  means  of  the  initial 
conditions  (120).  It  will  be  found  that  only  the  first  power  of  5  is  needed, 
and  then  only  in  terms  independent  of  «;  elsewhere  it  will  be  omitted. 
Likewise  only  the  first  powers  of  a,  and  a4  will  be  needed,  and  therefore 
the  higher  powers  will  be  omitted.  Since  a,  and  o,  enter  only  in  the  com- 
binations a,+o!  and  a,+o,,  we  may  omit  them  for  brevity  until  the  end, 
using  simply  a,  and  a,  ,  and  then  restore  them  where  they  are  needed. 


192  PERIODIC    ORBITS. 

The  terms  of  degree  zero  in  e  satisfying  the  initial  conditions  (120)  are 


w<0)  =  a3e+p(1+5)T, 
<  =  (a2+ a2)e-<r'(1+5)T,  7/<°>  =  a4e-pa+5)T, 

c 


(123) 


where  i  =  V  —  1. 

The  terms  of  the  first  degree  in  e  are  defined  by  the  equations 


-F\    ,  --/<»-ffcB[  ]+*•{  f, 


i    i 


,      (124) 


where 


4(m<r  —  np)  '  2  (mp  +  no-)  ' 


+2  (2  -  mm')  a,  a,  e+w+»T+2  (2  +mm')  ax  a 


+2  (2+mm')  a2a3  e(-ffl+")r+2  (2  -mm')  a2a4  e(-<rl-p)T]  , 

\  \  =  \nia\  ezffir  -m'aj  e-2ff'T  +  (m'+m)  at  a3  e(<rt+»T+  (ni-m)  a,  a4  e("-p)r 
-  (m'-m)  a2a3e(-<ri+p>T-  (m'+m)  a2  a4  e<-"-'"-  }  . 


>  325) 


The  solutions  of  (1  24)  are  the  respective  complementary  functions, 
#{VlT,  K?e-aiT,  K?epT,  K^c~pr,  plus  terms  of  the  same  character  as  their 
right  members.  In  the  solution  of  the  first  equation,  the  coefficients  of 
these  terms  are  respectively  the  coefficients  of  the  right  members  written  in 
the  order  given  in  (1  25),  omitting  the  term  cos2vTr,  divided  by 

—  ffi,      -\-tri,       —3cri,       +p,       —  p,       —  2<n'+p,       —  2trt  —  p. 
The  term-KwiSic*/2A)  cos2\/Tr  gives  rise  to 


The  corresponding  divisors  in  the  solution  of  the  second  equation  are 
respectively 

-{-ffi,        +3ot,        —  <n,        2o-i+p,        2<ri  —  p,        +p,        —  p, 
and  the  terms  coming  from  (+m-Z?t'c2/2.A)cos2\/jT  are 

mEic? 


-.x  .    . 

2(4^4.  — 


<>^<  II.I,.VMN<;    SATKI.LITKS       HUM     Ml;  I  MOD. 


193 


III  tin1  solution  of  tin-  third  and  fourth  equation-  it  i-  uiinece.-sary  to 
compute  the  terms  which  carry  a  and  a,  as  factors.  Omitting  these  term-. 
the  divisors  for  the  third  equation  arc  respectively  -  p,  '2<ri  —  p,  —  2<ri—p, 
and  the  term-  coming  from  (  —  nEc/'2A)  cos2  v.l  T  are 


'• 


For  the  fourth  equation  the  corresponding  quantities  are 

.....  •'rt 

The  solution  of  the  la>t  e<|Uation  of  i  124)  is 


I 
• 


2(p*+2tpVT) 


r    i 


I 


3BC04J 


(126) 


The  constants  of  integration  K[l}  .  .  .  .  ,  A'J"  are  determinod  by  the 
conditions  that  7/J°  ,  ....  »"'  shall  vanish  at  T  =  0,  and  the  constants 
LI"  and  /,;"  by  the  conditions  2,(0)=*;(0)  =  0. 

It  is  necessary  to  compute  all  non-periodic  terms  of  »,  ,  it,  ,  and  2  which 
are  of  the  second  degree  in  t  and  which  are  independent  of  a,  ,  .  .  .  ,  o4  , 
and  6.  The  right  members  of  the  differential  equations  involve 


- 


(127) 


The  quantities  j-,  and  T/,  are  defined  by 

X,  =  Mj"  +  Mi"  +  Mi"  +  Mi",  I/,  =  Hi(  Mi"  -  Mi")  +  m(U?  -  Ui"). 

In  order  to  get  all  the  non-periodic  parts  of  the  solutions  at  this  >tep,  the 
terms  of  the  differential  equations  which  are  non-periodic,  that  is,  which 
carry  p  in  the  exponential,  must  be  retained;  in  the  first  and  second 
equations  the  terms  in  r"T  and  c~"T  respectively  must  be  retained;  and  in 
the  2-equation  the  terms  in  cos  V^T  and  sin  \TA T  must  l)e  retained,  for 
the.-e  |>eriodic  terms  give  rise  to  terms  in  the  solution  which  are  multiplied 
by  T,  and  which  therefore  are  not  periodic. 


194  PERIODIC   ORBITS. 

The  conditions  for  a  periodic  solution  with  the  period  -&•  =  -v*  =  T  are 

a         VA 

0  =  ut(T]  -  ut  (0)  =  [<>  (T)  -  u™  (0)]  +  [u?(T)  -  M<n>(0)]  e 

+   •   •   •  (i  =  l  -----  4), 


0=   z(T)- 


+  [22(71)-z2(0)]e2+ 
By  means  of  the  steps  explained  on  pages  191  to  193,  we  find  explicitly 


tC(0)  =  (a2+a2)[e-2T')(1+5)<-  1],     ^"'(T1)  -<(0)  =  a4[ 
z0(T)-   20(0)  = 


(129) 


M<»  (j1)  -wa)(0)  =  I  [  2mEi(2  —  mm')  -  F(  m'  +  w)  ]  at  a3 


[—  2mEi(2 


«">  (T\     ,,M(C\\  - 


if  [—  2mEi('2  +  mni)  +  F  (ni  —  m)]a1a4 
I  P 

[  —  2mEi(  2  —  mni)  —F(ni  +  m)]  a2a4  "I  r  -2^      1  ~\ 
2ffi  +  p  ~\\-e 

—  2mEi(  2  —  mni)  —F(ni  +  m)]  a^ 


. 


—  2mEi(  2  +  mni)  —F(ni  —  m)]  a^ 
2(rt-p 

_,    [  -|-  2mEi(2  —  mni  )  —  F  (  m  +  w&  )  ]  Q2a4  1  f  -2jrpT  _  i  "I 

um(T)  -wa>(0)  =  (  [-rc-ff^+n'Q-nFtK  +  2n^(2-n2)a1a, 

[n#(2  +  n2)  -  nFt]  a2  2n^c2      \  r  +2^  _  1  -i 

2  "     4  2         L6 


2  o-{  +  p  p 

[-nE(  2+n*)-nFi]c£         tnEc*    \  r  -*r»f  _  ,  i 
2cri-p  "  (4Z+7H  L6 


OSCILLATING   SATELLITES  —  FIRST   METHOD.  195 


. 
2<n  —  p 

n'Ko,  .  (nE(2  +  n*)  -  nF  i]aj 


(4A+p 


>     If  a.       _g« 
*)pJlp       2<rt- 


2<rt'  +  p  p 


, 

2<rt-p 

x  [o-^  -i]  +2m' 


a, 


-  n^a.q,  .  [nE(2  +  n^)  -  nFi]dj 

p 

"-   n 

, 


2<rt'  +  P 

,   [-n^(2  +  n>)-nft]a;         2n^    If  a.          a, 

2<rt-p  "  (4A  +  p*)pJ  Ip       2<rt 

x 


—  p 


,  -l^, 

p 

2nEc>        fa,  . 

2<n-P 

-* 


p 

2nEct 


t     1  f  a,  .       a,     ] 
1)pJ  IP       2«rt  +  pJ 


x  fe-^  _  fl  _  nFi 


2<rt  —  p 


,  2nE(2-n1)alat,[nE(2+n*)-nFi]al        2nEc*    } 
p  2fft+p  (4^+p')pJ 

K  +  _4L_}r6^  -ll  +nFi  l(»E(2  +  n?-nFi]a\ 
IP      2<n-pJ«-  2<n+p 


,  2nE(2-nt)alat  ,  [-n 


p  2<n  —  p 

«» 


196  PERIODIC    ORBITS. 


where 


L  -  1  mETi  ft2  mE  (2 


n2) 


3(7 


2mn~ETi 


o-p  3<r 

2)  +  2nF] 


,  _     .  „„,. 

O       I  n  I  't     i.       JL.     L 

2       2  J 


, 


]1 
J 


o-p  J  B 

ZnFCTi       3 


M= 


~2B~ 

r)        smBETi 


<rpA~  (4A-<r*)A  apA 

mECTi      nFCTi 
AB  4AB 


(130) 


There  are  corresponding  equations  for  u™(T)—u™(0)  and  22(T)— z2(0). 

The  third  and  fourth  equations  of  (128)  can  be  solved  for  a3  and  a4  as 
power  series  in  f,  alt  a2 ,  and  5,  vanishing  with  e.     The  explicit  results  are 

-n2)  +  nFi]  a\_2nE(2-  n2)  a^ 
—P  P 

[—     'M    fl       ( O         [        -VJ^X  |         fy\    ft    n   I   y|2 

+                     /&-C/    ^  ^^  IV    )     n~    ///^    tj  tin        r  ^   f  tiJV 
-f-  .         — ^^c-^- 


\[ 
4      1 


\a\  _  2nE(2-ni)a1at 


_ 
20-t-fp  p 

,   [+nE(2+ri>)+nFi]al  , 

h2€ 


(131) 


When  these  expressions  for  a,  and  a4  are  substituted  in  the  first,  second,  and 
fifth  equations  of  (128),  we  obtain 

0  =  (a1  +  a1)[e+w5'-l]+(a1  +  a1)[(a1  +  a1)(a2  +  a2)L1  +  c2M1]€2+   •  •  • 


-mo- 


4A-a2 
3(7 


] 


(132) 


OSCILLATING    SATELLITES—  FIRST   METHOD.  197 

After  removing  the  factor  <    from  the  hist  equation,  solving  for  6,  and 
substituting  tin-  result  in  the  first  two  equations.  \ve  liave 


where 

j        L>  "/>'  3C(4-n') 

'  '  '      ~~ 


rmp+nff_    i  j/   —mo  np      \~  _     9C 

ap 


M  -  ff_    i      /   — 

•         L       ap  ' 


0  =  (a,  +  a,M(n,  +  a,  )(«,  +  at)L+c'J/]  «'+   •  •  •  , 


(134) 


Kquation-     i:>3)  can  not  be  solved  for  o,  and  a,  as  power  .-erics  in  t, 
vanishing  with  t.  unless 

,U","^+cMf]=0,  at[a,o,L+c1M]  =  0.  (135) 

One  solution  of  these  equations  is  aj  =  a,  =  0,  and  with  this  determination  of 
a  and  o,,  equations  (133)  are  uniquely  solvable  fora,  and  a,  as  power  series 
in  (.  vanishing  with  t.  In  this  case  the  generating  solution  reduces  to  the 
form  of  that  of  Class  A.  But  the  orbits  of  Class  A  heretofore  treated  were 
those  for  which  V  A  and  a  are  incommensurable.  Thi-  restriction  was  not 
necessary  in  order  to  prove  that  orbits  exist  which  re-enter  after  one  revo- 
lution, but  it  was  not  certain  that  there  are  not  others  re-entering  only  alter 
many  revolutions.  The  uniqueness  of  the  solution  of  (133),  for  a,  =  r/,  =  0, 
proves  that  all  of  the  orbits  of  ('/</.-;*  A,  of  the  analytic  ti/f>e  under  consideration 
In  />  .  1  1  -enter  after  a  single  revolution. 

At  the  beginning  of  the  present  discussion  the  assumption  was  made 
that  c  is  distinct  from  zero,  and  this  permitted  the  suppression  of  the  last 
equation  of  (122).  If  we  had  assumed  that  a,  is  distinct  from  zero,  we 
could  have  suppressed  the  first  equation  of  (122);  and  solving  in  a  different 
order,  we  should  finally  have  arrived  at  two  equations  corresponding  to  (,133) 
containing  (c-r"y)  asa  factor.  The  equations  would  have  been  found  solvable 
after  imposing  the  condition  c  =  0,  and  we  should  have  arrived  at  the  eon 
elusion  that  all  orbits  of  Class  B  re-enter  after  one  revolution. 

Equations  (135)  also  have  the  solution 

a,a,L+c'M  =  0.  (136) 

This  equation  defines  <•  when  a,  and  a,  have  been  given  arbitrary  values. 
If  the  orbits  are  to  be  real,  a,  and  a,  must  be  conjugate  complex  quantities. 
Under  these  circumstances  their  product  is  positive,  and  L  and  M  must  be 
opposite  iii  sign  in  order  that  c  shall  be  real. 

After  the  condition  (136)  has  been  applied,  equations  (133)  become 

«[a1  ,  a.  ,  t]+   •   •   •   , 
-«[o1  ,  a,  ,  e]+   •   •   •   • 


198  PERIODIC    ORBITS. 

Since  the  terms  of  these  equations  which  are  independent  of  e  are  identical, 
except  for  the  non-vanishing  factors  e^-fa,  and  a-j+Oj  ,  it  follows  that  if 
one  is  solved  for  ax  and  the  result  substituted  in  the  other,  the  latter  becomes 
divisible  by  e.  After  dividing  out  e,  there  is  a  term  independent  of  a.,  and 
e  which  must  be  equal  to  zero  in  order  that  the  equation  may  be  solved 
for  Oj  as  a  power  series  in  e,  vanishing  with  e.  This  term  involves  the 
coefficients  of  e3  in  the  original  solutions  (122),  since  e3  has  been  divided 
out.  Likewise,  terms  enter  from  lower  powers  of  e  through  the  elimination 
of  a3  ,  a4  ,  and  5.  It  is  not  possible  to  construct  these  terms  without  an 
unreasonable  amount  of  work.  But  we  see  from  the  way  in  which  they 
originate  that  they  are  homogeneous  of  the  fourth  degree  in  a:  and  «2. 
Unless  one  or  the  other  of  these  constants  is  absent,  their  ratio  is  determined 
by  this  constant  term  set  equal  to  zero.  If  one  is  absent,  the  only  solution 
is  the  other  set  equal  to  zero,  which  throws  us  back  on  Class  A,  which 
has  been  already  completely  treated. 

Suppose  both  constants  are  present  and  that  their  ratio  is  determined. 
Since  they  must  be  conjugate  in  order  that  the  orbit  may  be  real,  the  solu- 
tion for  the  ratio  has  the  form 


at  "  a-b  V=l  '  a2+62 

It  is  clear  that  only  for  special  values  of  the  coefficients,  which  might  never 
be  possible  in  the  problem,  could  the  solution  for  the  ratio  have  this  form. 
The  complexity  of  the  problem  is  such  that  no  further  attempt  will  be  made 
here  to  determine  whether  there  exist  solutions  of  Class  C  which  are  dis- 
tinct from  those  of  Class  A  and  Class  B. 

If  the  attempt  is  made  to  construct  the  periodic  solution  of  which 
(32)  are  the  terms  independent  of  e,  no  difficulty  will  be  encountered  until 
the  terms  in  e2  are  reached.  Then  it  will  be  found  that  equations  (135) 
must  be  satisfied  in  order  that  the  solutions  at  this  step  shall  be  periodic. 
That  is,  step  by  step,  the  construction  agrees  with  the  existence,  though 
the  computation  is  somewhat  less  laborious. 


CHAPTER  VI. 

OSCILLATING  SATELLITES. 
SECOND  METHOD.* 

97.  Outline  of  Method. — The  problem  treated  in  this  chapter  is  the 
same  as  that  considered  in  the  preceding,  but  the  method  employed  is  quite 
different.     In  this  particular  question  the  preceding  method  is  somewhat 
more  convenient,  but  in  other  problems  where  the  same  general  style  of 
analysis  can  bo  used  it  is  much  less  so. 

There  is  a  definite  physical  situation  for  which  the  analysis  is  to  be 
develo|>ed.  One  of  its  principal  features  is  that  the  periodic  orbits  form  a 
continuous  series  from  those  of  zero  dimensions  at  the  points  of  equilibrium, 
and  as  they  vary  in  dimensions  the  periods  undergo  corresponding  changes. 
In  the  analysis  of  Chapter  V  the  dimensions  were  controlled  by  means  of 
the  scale  factor  «'  and  the  varying  periods  were  properly  secured  by  the 
introduction  of  5  and  its  subsequent  determination  in  terms  of  «'  As  «' 
approached  zero  the  orbits  approached  zero  dimensions  and  the  period 
approached  the  value  which  corresponds  to  6  =  0. 

In  the  present  treatment  no  parameters  corresponding  to  e'  and  6  are 
employed.  Instead,  we  introduce  a  parameter  X  by  means  of  /*  =  /*0+X, 
where  MO  is  kept  fixed  in  numerical  value  while  X  is  a  parameter  in  terms  of 
which  the  solutions  are  expressed.  Periodic  solutions  are  found  for  all  X 
whose  moduli  are  sufficiently  small,  but  only  those  solutions  belong  to  the 
physical  problem  for  which  X  =  M~MO-  The  dimensions  of  the  physical  orbit 
depends  upon  this  value  of  X,  and  its  period  depends  upon  MO-  That  is,  we 
find  a  family  of  periodic  solutions  having  a  constant  period  depending  upon 
M, ,  but  only  one  of  them  belongs  to  the  physical  problem.  It  is  because  of 
this  fact  that  it  is  not  necessary  to  make  the  period  variable  and  dependent 
upon  the  parameter  in  terms  of  which  the  solutions  are  developed. 

98.  The  Differential  Equations.— We  shall  start  from  equations  (6) 
of  Chapter  V,  omitting  the  accents  which  will  not  be  needed.     The  right 
members  of  these  equations  involve  the  parameter  n  explicitly  in  the  last 
two  terms  of  U,  and  implicitly  through  r,  and  r,  which  depend  upon  rfM)  de- 
fined in  (4).     We  shall  make  the  transformation 

M  =  M,+X,  (1) 

but  it  is  not  necessary  to  do  so  in  all  places,  both  explicit  and  implicit,  in 
which  this  parameter  occurs  in  the  differential  equations. 

The  problem  of  oscillating  satellites  was  first  treated  by  the  author  by  the  method*  of  this  chapter. 
However,  the  two  methods  were  reported  on  simultaneously  in  the  paper  referred  to  at  the  beginning  of 
Chapirr  V.  «» 


200  PERIODIC    ORBITS. 

For  simplicity  the  transformation  will  be  made  where  it  appears  explicitly 
in  U,  and  elsewhere  M  will  be  supposed  to  retain  its  original  given  value, 
which  is  regarded  as  a  fixed  constant.  This  particular  generalization  of  the 
parameter  ju  is  not  the  only  possible  one,  and  the  series  obtained  differ 
according  to  the  particular  generalization  made,  but  when  the  conditions 
for  convergence  are  satisfied  their  sums  are  identical  in  t. 

After  the  transformation  (1),  equations  (6)  of  Chapter  V  become 

x*-2y'-(l+2AJx=    P,(x,  if,  z%  X), 

2  °        \  \ 

x,  y  >  z~,  A;, 

2  2        "\  "\ 

xi  y  >  z  )  xj, 


(2) 

y" -\-   A      y  —  y  P  (v     ifl      y2      \} 

r  YIO  z  —  zr2\j.,  y  ,  z  ,   A^, 
where 


A    .  .         Mo    ,    _Mp_ 

•**•  r(0)3        I     r(0)3 


r 
' 


it _  _i T)   +1       Mo  +  _Mo  r<   i    •        Mo    i     Mo 

"     "  r(0)3     I     r(0)3  '  **•          +      7-(0)4     —  r(O)4  '  ^8          T       r(0)5      T    ..(0)5  ' 

'1  '2  —      'l  '!  '2  '2 


(3) 


the  signs  in  B0  being  the  first,  second,  or  third  according  as  orbits  in  the 
vicinity  of  (a),  (6),  or  (c)  are  in  question.  The  regions  of  convergence  of 
P!  and  P2  are  precisely  the  same  as  those  found  in  §77. 

We  shall  need  the  differential  equations  in  the  normal  form  so  far  as  the 
linear  terms  are  concerned.  When  the  right  members  of  (2)  are  put  equal 
to  zero,  their  solutions  are 


(4) 


where  Kl}  .  .  .  ,  7v4  ,  c,  ,  and  c2  are  arbitrary  constants  of  integration,  and 
where  +  a0  V—  i,  —  <r0  V  —  l,  +  p0  ,  and  —  pa  are  the  four  roots  of 


x  =    ,  e' 

» (K3  e"°' - 


and  where  also 

_  a2+.0  _      --0 

°~     ~  °~ 


Qonsequently  the  normal  form  is  secured  by  the  transformation 

.r  =+(i/,+O  +  («3+iO, 

x'  =  +  <r0  v'-^T  (M,  -  ;/,)  +  pcl  (  it,  -  1/4)  , 

_  .  \    (6) 

y  =+n0V-i(u1-u. 


osrn.LATINCJ    SATELLITES — SECOND    MKIIKH).  201 

which  reduces  equations  (2)  to 

-Nopo)  V—l 


IM  =  -  ...  ,  f,  X) 

-'  »/offC-n«p«)V-l         2(ro,p»+n«<r,) 

n.P.foy'.z'.X)      .     yPt(x,  y*t  *,  X)  (7) 

2(m,<r0-n«po)  2(m«p,+n«<ro) 

»P.  (x,  y»,  z»,  X)  yP  (a,  y*,  z»,  X) 

2(m.  *.-**) 

,(x,  y1,  *»,  X), 
when-  /=  v^T  and  where  P,  and  P,  have  the  values  given  in  (3). 

99.  Integration  of  the  Differential  Equations.—  Equations  (2)  admit 

the  integral 

!/n+z"-i*+yt+At(2i*-yt-z>)+A'(2x>-yt-z')\ 

(8) 


1 
| 


=constant, 


which  holds  for  x,  y,  and  z  within  the  region  for  which  the  series  converge. 

Since  there  is  always  a  component  of  acceleration  toward  the  xj/-plane, 
there  can  be  no  closed  orbit  entirely  on  one  side  of  this  plane.  Therefore,,  in 
all  cases  we  can  take  the  origin  of  time  so  that  z  =  0  at  t  =  0.  Suppose 

Uj  =  aj,          z  =  0,          z'  =  y   aU  =  0.  (9) 

We  now  integrate  equations  (7)  as  power  series  in  the  an  y,  and  X.      The 
solutions  arc 


.  .  .  ,  a4,  y,  X;  0, 

*'  +P*(«I>  •  •  •  ,  «4,  y,  *;  0, 

.  .  .  ,  o4,  y,  X;  0,  ,JQ, 

i»  •  •  •  •  <i4,  y,  X;  0, 

i>  •  •  •  >  o4,  y,  X;  0, 

a     -v   X-  /") 
i>  •  •  •  >  a4»  it  A»  *y» 

where  p,,  .  .  .  ,  p,  are  power  series  in  o,,  .  .  .  ,  o4,  y,  and  X.  The 
moduli  of  these  parameters  can  be  taken  so  small  that  the  series  converge 
for  all  O^J^T,  where  T  (finite)  is  taken  arbitrarily  in  advance  (§16). 
The  j)j  arc  of  the  second  and  higher  degrees  in  the  a},  y,  and  X.  It  follows 
from  the  way  in  which  X  was  introduced  that  the  p,  identically  vanish 
for  a,=  •  •  •  =a4  =  -y  =  0.  Since  the  last  equation  of  (7)  contains  2  as  a 
factor,  pt  =  p,  =  0  for  7  =  0,  whatever  the  other  initial  conditions  may  be. 


202  PERIODIC    ORBITS. 

100.  Existence  of  Periodic  Solutions. — Since  the  right  members  of 
equation  (7)  do  not  contain  t  explicitly,  sufficient  conditions  that  (10)  shall 
be  a  periodic  solution  with  the  period  T  are 


'""  —  ij     -TVs\i)—  /A,W> 

-"'T      -l]   +p4(T)-p4(0), 

0=  z(T)-  z  (0)  =  -^=  sin  VT0 T  +  ps(T)-p,(0), 
0=z'(T)-z'(0)=7[cosVT0T-l]+p6(T)-p6(0). 

The  last  two  equations  of  (11)  are  satisfied  by  7  =  0.  Suppose  7^0.  Then 
it  follows  from  the  form  of  the  integral  (8)  that  unless  VZ0T  =  (2n-}-l)ir/2, 
where  n  is  an  integer,  the  last  equation  is  a  consequence  of  the  first  five. 
We  shall  suppose  T  does  not  have  one  of  these  special  values,  and  we  shall 
suppress  the  last  equation  since  it  is  a  redundant  condition.  The  first 
five  equations  are  to  be  solved  for  Oj ,  .  .  .  ,  a4 ,  and  7  in  terms  of  X,  and 
we  can  use  only  those  solutions  which  vanish  with  X .  These  equations  are 
satisfied  by  e^  =  •  •  •  =  a4  =  7  =  0.  In  order  that  this  may  be  not  the  only 
solution  vanishing  with  X,  the  determinant  of  the  coefficients  of  the  linear 
terms  in  a, ,  .  .  .  ,  a4 ,  and  7  must  be  zero.  This  condition  is  explicitly 


_  j-j  |y.T  _  ^  |-e-,.T  _  j  J 

This  equation  has  the  solutions 

T>  =          '  T'  =  <"  an  integer).          (13) 


0 

Consider  first  the  solution  T  =  Tj  .  For  this  value  of  T  the  determinant 
of  the  linear  terms  in  ai  ,  .  .  .  ,  a4  of  the  first  four  equations  of  (11)  is  distinct 
from  zero  unless  v<ra/2  VT<>  is  an  integer.  This  condition  can  not  be  fulfilled 
for  all  v  unless  <r0  is  an  integral  multiple  of  2  V~A0  .  Now  since  o>  =  <TO  V—l 
satisfies  w<+(2-AV+(l+2,40)(l-,40)=0, 

this  condition  can  not  be  fulfilled  unless  A0  is  negative,  but  from  its 
definition  in  (3),  A0  is  always  positive.  Therefore  there  are  values  of  v  for 
which  vaa/2  VA~<,  is  not  an  integer,  and  one  of  these  values  of  v  is  necessarily 
unity.  It  follows  that  the  first  four  equations  of  (11)  can  be  solved  for 
au  .  .  .  ,  a4  as  power  series  in  7  and  X.  Since  ult  .  .  .  ,  ut,  and  z  vanish 
with  ^  =  •  •  •  =  a4  =  7  =  0,  these  solutions  vanish  with  7,  and  since  the 
first  four  equations  of  (7)  are  functions  of  z2,  they  carry  72  as  a  factor. 
On  substituting  the  solutions  of  the  first  four  equations  of  (11)  in  the  fifth, 
it  becomes  a  power  series  in  7  and  X  alone,  and  is  divisible  by  7. 


OSCILLATING    SATELLITES — SECOND    METHOD. 


203 


In  order  to  prove  the  possibility  of  the  solution  of  the  fifth  equation 
for  y  in  terms  of  X,  and  to  determine  the  character  of  the  solution,  we  must 
work  out  the  first  terms  of  the  series.  Terms  in  X  alone  can  not  he  intro- 
duced from  the  solutions  of  the  first  four  equations  of  (10)  unless  the  fifth 
equation  is  divisible  by  7',  for  the  former  carry  7'  as  a  factor.  We  shall 
show  first  that  the  fifth  equation  of  (11)  has  a  term  in  y\. 

It  is  seen  from  (7)  and  (3)  that  the  coefficient  of  -yX  in  the  expression 
for  z  is  defined  by 

(14) 


The  solution  of  this  equation  satisfying  the  conditions  that  z,,(0)  =  0  and 

*;,«>)  =o  is 

(15) 


It  follows  from  the  non-periodic  term  of  this  equation  that  the  fifth  equation 
of  (1  1)  has  a  term  in  -yX,  and  therefore  that  the  solutions  exist.     To  get  their 
character  we  must  find  the  terms  of  lowest  degree  in  y  alone. 
The  coefficients  of  7*  are  defined  by 


a.o) 


+  <rX''0>=- 


4  | 

~ST   ' 


0.0) : 


where 
[ 


(16) 


la-  »» 


4-  ?/    -I-  2*  1 
o  T  tfi.o  i  zi.oJ> 


y,0.        (17) 
'  as  a  factor. 


We  shall   need  only  the  terms  M"|W,  .  .  .  ,  u^n  carrying  7 
Hence  in  the  first  four  equations  we  may  omit  z,,,  and  y,,0 .     From  equa- 
tions (6)  and  (10)  we  have 


2VA,  V-i 

a*v 
2VTt  v^^T 

^"Y 

t  v/:rT 

047 

•  v/Trr 


^i«  "I 


(18) 


204  PERIODIC    ORBITS. 

Therefore,  integrating  (16)  so  far  as  the  first  four  equations  depend  upon 
terms  involving  72  as  a  factor,  and  determining  the  constants  of  integration 
so  that  w?'0),  z,.0,  and  z2.0  are  zero  at  t  =  0,  we  get 

#0)  Bm2VTtt, 


w  .    =  a.  .      -  '  0> 


uf-  0)  =  a'2'  0)  +  a,*  0)  e+p"      +  a£  0)  cos  2  V3;  <  +  6«-  »  sin  2  v^  <  , 
M«.  °>  =  a*  0)  4-  a*  0)  e-"°'      +  a»-  0)  cos  2  Vi;  «  -  6<22'  0)  sin  2  V3;  <  , 


" 


2  (o-o  —  2  V^0)<r0  V^0  V-l       2  (  <r0  —  2 

a2  7  e(-ff°-  v/^v/r"1  '  a3  7 

' 


"  "  '^I     2(p«+2 


2(p0—2VAf>V—l)p0VA0V- 
a47 


2(p0+2  V^"0  V-l)p0 

where 

r/(2,o>_   ,  3  mo  Bo  7s  ,  a«.o.=  +  3  n0  .Bo  72 

a'«  8  (m,  (^0-n0  p0)  A0  a0  8  (?/?0  o-0-r?0  Po)  A0  Po 

(20)_  3  W0  BO 

a'2'          h 


(19) 


(2  o>  =  3m0  Bo  72  ,    _«.  o)  =  _  _3«o  Bo  72 

(72)<rn  2(w?0ff0-n0p0)  (4  ,40 


(2,0)_  3  W0  Bo 

h  - 


(20) 


The  terms  of  z  of  the  third  degree  in  7  are  defined  by 

3B0x2,o2l,0+C04o.  (21) 


We  shall  need  only  the  non-periodic  terms  and  those  whose  period  is  not 
2ir/V^.  The  former  come  from  terms  in  sin  V~A~0t  and  e*p°',  and  the  latter 
from  terms  involving  e*"-^"  ,  which  together  with  e  *pa<  are  introduced  by 
x,  o  Since 


"-'   II. LATIN*;    SVTKI.LMl.-       OOONO    MKTH*H).  'JO.") 

we  tind  for  the  rei|iiire»l  terms 


sin 

'(22) 


. 

2    «    .o  =      gt   sin 


Thi-refore  the  part  of  the  solution  of  (21)  not  having  the  period  T,  is 


|23) 


\\'c  can  now  write  the  conditions  for  the  existence  of  periodic  solutions. 
t'pon  using  the  results  just  obtained,  we  find  for  the  values  of  (11) 


trirg.VZj 
'  **  -  O- 


rrp, 


,7.  [(  -  »•  .^  -  1]  + 


-i 


(24) 


206  PERIODIC    ORBITS. 

Upon  solving  the  first  four  equations  of  (24)  for  ou  .  .  .  ,  a4,  substituting 
the  result  in  the  last,  and  reducing  by  (20),  we  get 


virA'y\  _    SvirBl  (  m0po  +  n«  g0  )  y 


3 


. 
'  ' 


_ 

-  oj)  (4A0  +  p°)  A\ 


,   _  QvirBl  (  Wopo+Woffo)  (T<,p0y3  o       , 

---  ' 


From  the  characteristic  equation  and  equations  (5),  we  find 
m0 (r0  —  W0p0=  —  (1  +  2.A0)  — ^    —  >          m0p04-w0(r0=  -  - — -> 

2ff0po  (25) 

Hence,  after  dividing  by  y,  we  have 

'   2+  •  •  •  ,        (26) 


_ 

-  7  A0+  ISA?) 

which  can  be  solved  for  y  in  terms  of  X  in  the  form 

.  (27) 


Upon  substituting  this  result  in  the  series  for  e^  ,  .  .  .  ,  a4  when  they  are 
expressed  in  terms  of  y  and  X  from  (24),  we  have 

a,  =  XP,(X»)  (i  =  l,  ...,4).  (28) 

After  (27)  and  (28)  are  substituted  in  (10)  the  coordinates  ut  and  z 
become  power  series  in  X*,  vanishing  with  X1,  and  they  are  periodic,  since 
the  conditions  for  periodicity  have  been  satisfied.  The  series  converge  for 
|X|  sufficiently  small.  The  radius  of  convergence  depends  on  M  and  MO  and 
it  is  easy  to  see  from  the  explicit  forms  of  the  equations  that  it  remains  finite 
as  /z0  approaches  /x.  For  X  =  /x—  fj^  the  orbits  belong  to  the  physical  problem, 
and  /xc  can  be  taken  so  near  p  that  the  series  converge.  That  is,  periodic 
solutions  exist  having  the  form 

x=  S  z^,         y-  2  y^,         z=  i  z,X%  (29) 

1-1  <-  1  <=i 

where  the  xt,  y(,  and  z,  separately  are  periodic  functions  of  t  having  the 
period  2ir/VAa  . 


OSCILLATING    SATELLITES — SECOND    METHOD.  207 


I 


The  last  t\\o  equations  of  (11)  arc  satisfied  byy  =  0.     For  T  =  2ir/V/l 
the  determinant  of  the  linear  terms  of  the  first  four  equations  is  distinct 

from  zero;  therefore  their  only  solution  for  a ,  at  as  power  scries  in 

X,  vanishing  with  X,  is  o,=  •  •  •  =a4  =  0.  Hut  then  M,  ,  .  .  .  ,  ut,  and  z 
are  identically  zero.  That  is,  the  solutions  having  the  period  2r/VA9  are 
in  three  dimensions  and  not  in  two  alone.  In  this  respect  they  agree  with 
the  solutions  of  Class  A  of  Chapter  V.  It  will  now  be  shown  that  for 


these  solutions  are  those  of  Class  A. 

The  solutions  of  Class  A  were  developed  as  power  series  in  «'  of  the 
form 

z=   S  V,  y=   2  y,t",  z=   S  z<tn,  (30) 


1-1 


where  xt,  y,,  and  z^  are  periodic  in  T  =  t/(l  +  5)  with  the  period  2ir/vT.  The 
At  appearing  in  the  xt,  ylt  z,  of  (29)  is  different  from  the  A  of  (30);  in  the 
former  n  has  been  replaced  by  ^  in  certain  places,  while  in  the  latter  it 
remains  n.  Now  let  /z  =  /^+X  in  (30)  in  those  places  where  this  transforma- 
tion was  made  in  the  development  of  (29),  and  develop  the  right  members  as 
power  series  in  X.  The  period  of  the  solutions  (30)  when  expressed  in  t  is 
2ir(l+d)/V~A,  where  5  is  a  known  power  series  in  «'.  Make  the  transfor- 
mation on  n  in  this  expression  so  that  A  =A9+p  (X)  and  set  it  equal  to  the 
period  of  (29),  viz.  2H-/V7,.  Since  S  starts  with  a  term  of  the  second  degree 
in  «'  this  equation  determines  «'  as  a  power  series  in  X*.  Substituting  this 
expression  in  (30),  we  have  these  equations  expressed  as  power  series  in  X*. 
For  sufficiently  small  |  X  |  these  series  converge,  the  coefficients  of  each 
power  of  X*  separately  are  periodic  with  the  period  2r/\/A9,  they  identically 
satisfy  the  correspondingly  transformed  differential  equations,  and  they  are 
identically  equal  (in  t)  to  (30)  in  their  original  form.  Having  the  same  form 
as  (29),  it  follows  from  the  uniqueness  of  these  solutions  that  they  are 
identical  with  them.  That  is,  equations  (29)  and  (30)  are  two  different 
sets  of  expressions  for  the  coordinates  of  the  orbits  of  Class  A. 

Now  return  to  the  consideration  of  equations  (11).  Let  the  last  two 
be  satisfied  by  y  =  0.  As  we  have  seen,  there  is  no  solution  of  the  first  four 
vanishing  with  X=0  except  a,=  •  •  •  a4-0  if  T  =  TI=2T/V/T0.  Therefore 
we  take  T  =  T,  =  2ir/<r0 .  The  first  equation  is  redundant  because  of  the 
existence  of  the  integral  (8),  and  will  be  suppressed.  The  third  and  fourth 
equations  can  be  solved  for  a,  and  a,  as  power  series  in  a, ,  a, ,  and  X, 
vanishing  with  at  and  a, .  When  these  results  are  substituted  in  the 
second  equation,  we  have 

P(oIfaf,X)-0,  (31) 

where  P  identically  vanishes  with  a,  and  o, . 


208  PERIODIC    ORBITS. 

We  have  one  equation  for  the  determination  of  the  parameters  Oj  and 
a2.  Consequently  we  may  impose  one  condition  upon  them.  It  will  be 
convenient  to  take  ax  and  a2  so  that 

x'  =  OaU  =  0,  (32) 

a  condition  which  is  satisfied  in  every  closed  orbit  for  which  the  right  mem- 
bers of  (2)  converge.     From  equations  (6)  it  follows  that  this  condition  is 

cr0V^T  (a,-a2)+p0(a3-a4)=0.  (33) 

We  shall  regard  this  equation  as  a  relation  between  ^  and  a2  which  is  to  be 
used  in  connection  with  (31)  for  the  determination  of  these  parameters. 

In  order  to  complete  the  proof  of  the  existence  of  the  solutions  it  is 
necessary  to  discuss  the  second  and  third  degree  terms  of  (31).  It  follows 
from  (7)  and  (3)  that  the  terms  of  the  second  degree  are  determined  by 


(34) 

1(2) 


dU4         I  7,<2>  _      I      7,   f      1(2)        I      I        I  (2) 

~~jy~  TT    PoM4         i    "n    ]        i    i     i     > 
where 

2A'\xl+lB.[-24+ri\t         (        m       A'\i0ii         (35) 
— 


It  follows  from  the  forms  of  the  right  members  of  these  four  equations 
that  their  solutions  will  contain  Poisson  terms,*  whose  coefficients  involve 
Xttj  ,  .  .  .  ,  Xa4  respectively  as  factors.  The  coefficients  of  all  the  other 
terms  are  of  the  second  degree  in  a:  ,  .  .  .  ,  a4  and  linear  in  B0,  and  those 
which  are  not  periodic  involve  a3  or  a4  at  least  to  the  first  degree.  Conse- 
quently, when  we  solve  the  third  and  fourth  equations  of  (11)  for  a3  and  o,,, 
the  results  will  start  with  terms  of  the  second  degree  in  o,  ,  a2  ,  and  X.  When 
these  results  are  substituted  in  the  second  equation  of  (11),  it  will  contain 
a  term  in  X  Oj  and  terms  of  the  third  degree  as  the  lowest  in  a,  and  cu  alone. 
If  we  now  eliminate  ot  by  means  of  (33),  we  have  an  equation  whose  terms 
of  lowest  degree  are  a,X  and  a2  .  We  shall  verify  first  that  the  coefficient 
of  a2X  is  not  zero. 

*In  Celestial  Mechanics  terms  which  arc  of  the  form  of  I  multiplied  by  cosine  or  sine  terms  are  called 
Poisson  terms,  from  the  results  in  Poisson's  theorem  on  the  invariability  of  the  major  axes  of  the 
planetary  orbits. 


II.  I.  \  I  IN',    -\ll.l.l.lll.-     -SECOND    MKTHOD.  L'O'.t 

It  follows  from    .',1    and    36)  that  tin-  Poisson  terms  in  i/.,'  are 


^  _     > 


•  «•*    " 


V-l 
Hence,  so  far  a  term-  arc  concerned,  we  find 

5s-  -1. 

o  +  HflffoJ 


which,  I  iy  ('({nations  (25),  reduce.-  tu 

-  <«»  - 


Therefore  the  coefficient  of  Xo,  in  (31)  is  not  zero  unless  .!'  =  ().  But 
except  for  tli"  center  of  liliration  </»  when  the  finite  masses  are  equal,  and 
in  this  ca<e  a  different  generalization  of  the  parameter  »  can  be  made  to 
kee|>  it  distinct  from  zero.  Consequently,  since  all  the  equations  are  iden- 
tically satisfied  by  a,  =  •  •  •  =a,  =  0,  the  equation  obtained  by  eliminating 
a,  between  (31)  and  (33)  is  divisible  by  a,,  after  which  there  is  a  term 
in  X  alone  whose  coefficient  is  distinct  from  zero.  Therefore  the  equation 
can  be  solved  for  a,  in  terms  of  X,  vanishing  with  X,  and  the  periodic 
solutions  exist. 

The  form  of  the  solution  depends  upon  the  degree  of  the  term  of  the 
lowest  degree  in  a,  alone  in  the  final  equation  after  a,,  a,,  and  a,  are  elimi- 
nated. It  is  ea.-y  to  show  that  the  coefficient  of  oj  in  this  equation  is  not 
identically  zero.  It  has  been  shown  that  the  terms  arising  from  the  solu- 
tions of  (34)  involve  Bt  linearly.  There  are  also  terms  of  the  third  degree 
in  a,,  .  .  .  ,  a4  arising  from  the  terms  of  the  third  order.  The  terms  of 
«,  of  the  third  order  are  defined  by 

H    vA-  • 


dt 

where 


The  Poisson  terms  in  the  solution  of  (37)  involving  ('„  as  a  factor  are 

Mo)  =  _    3(7,  w,(2  -  np  a.aite-''^^1     ,     aC^n.  (4  -  3  np  a.  ajt  e-*<^  ' 
(mtfft—r^pt)  V—l  4(w«po+n«<T.)  V-l 

Hence  we  find  for  these  terms,  after  some  reductions, 

_t    /oo, 
.o,.   (38 


210  PERIODIC    ORBITS. 

It  follows  from  equations  (5)  that  -,  J-  of  (38)  does  not  identically  vanish. 
Hence  the  coefficient  of  a^  a\  in  the  expression  for  u2  (2w/ff3)  —  u2  (0)  consists 
of  terms  multiplied  by  B0  plus  non-vanishing  terms  multiplied  by  C0 . 

Now  suppose  at,  a3,  and  a4  are  eliminated  from  the  second  equation 
of  (11)  by  means  of  (33)  and  the  third  and  fourth  equations  of  (11).  It 
follows  from  the  properties  of  these  equations  that  in  the  result  the  term 
independent  of  X  and  of  lowest  degree  is  of  the  third  degree  in  a2  .  Its 
coefficient  consists  of  two  parts,  one  of  which  is  (38)  and  contains  C0  as  a 
factor,  while  the  other  contains  B0  as  a  factor.  If  the  coefficient  of  B0  is 
identically  zero,  the  coefficient  of  o|  is  distinct  from  zero,  because,  as  we  have 
seen,  the  part  involving  C0  is  distinct  from  zero.  Even  if  the  coefficient  of 
50  is  not  zero,  it  is  easy  to  show  that  the  sum  of  the  two  parts  of  the 
coefficient  of  o|  can  not  be  identically  zero  for  each  of  the  three  libration 
points  (a),  (6),  and  (c). 

Consider  the  points  (a)  and  (6).  The  quantities  A<>,  a0,  p0,  ma,  n0, 
and  (70  are  the  same  function  of  ^0  for  both  points,  but  B0  is  different  because 
of  the  change  of  sign  in  its  second  term  [eq.  (3)].  Consequently,  the  sum  of 
the  terms  in  B0  and  C0  can  not  be  identically  zero  in  MO  f°r  both  the  points 
of  libration  (a)  and  (6).  Hence  in  this  case  the  second,  third,  and  fourth 
equations  of  (11)  and  (33)  are  solvable  for  Oj ,  .  .  .  ,  a4  as  power  series 
in  X},  vanishing  with  X.  Therefore  the  periodic  solutions  with  the  period 
2ir/ff0  are  expansible  as  power  series  in  X}. 

In  a  manner  similar  to  that  used  to  prove  that  (29)  are  series  which 
represent  orbits  of  Class  A,  it  can  be  shown  that  the  orbits  now  under  con- 
sideration belong  to  Class  B. 


101.  Direct  Construction  of  the  Solutions  for  Class  A.  —  As  in  the 
method  of  Chapter  V,  the  coordinates  in  the  orbits  of  Class  A  are  most  con- 
veniently obtained  from  the  x,  y,  and  ^-equations.  Consequently  we  start 
from  equations  (2)  and  (3).  Since  the  solution  is  periodic  for  all  [X1]  suf- 
ficiently small,  each  term  of  the  expansion  separately  is  periodic  with  the 
period  2ir;  and  since  z  =  0  at  t  =  0,  each  term  in  the  expansion  of  z  separately 
vanishes  at  t  =  0. 

The  coefficients  of  X*  are  defined  by 

x';-2y'l-(l+2A0)xl  =  Q,      y^+2x'1-(l-A0)yi  =  0,      zl+A&  =  Q.  (39) 

The  solutions  of  these  equations  satisfying  the  periodicity  and  initial  con- 
ditions are 


2,  =  -       sinv/To  t,  (40) 

where  ct  is  so  far  undetermined. 


u;+2t't-(i-   .1    v.    :;/;  A 

3  B* 


MOLLATHra  SATELLITES— SECOND  METHOD.  211 

The  coefficients  of  ,\  :tre  defined  by 


(41) 


I  'pon  making  u>e  of  >  J(>>.  integrating.  and  applying  the  periodicity  and 
initial  conditions,  \\e  have 


3fi.C?  3g.(l  +  3;4.)c; 

4(1+2,40M0H     4(l-7i.+  l*4SM.  < 


whore  r,  is  so  far  arbitrary. 

It  will  be  necessary  to  carry  tho  computation  two  stops  further  in  ordor 
to  .-how  how  the  general  term  is  found.     Tho  coefficients  of  X1  arc  defined  by 


y't+2x'3-(l-    .40)»/,  =  0, 


„- 

|81 


..         .n 


(43) 


Consider  first  the  solution  of  the  third  equation.     In  order  that  it  shall  be 
periodic  tho  coefficient  of  sinv^f  must  be  zero,  or 


Ai  -«•  (44) 

Thi.-  <•( illation,  which  is  identical  with  (26)  of  the  existence  proof,  has  tho 
solutions 

. 2  V2A,Af 

C,  =  0,  C,  =  ±  -  •  / 1  e\ 

3 


212  PERIODIC    ORBITS. 

The  solution  ci  =  0  leads  to  the  trivial  case  x  =  y  =  z  =  0,  as  can  be  shown 
easily  by  an  induction  to  the  general  term.  The  double  sign  before  the 
radical  plays  the  same  role  as  the  double  sign  before  X!  in  the  existence. 
If  it  is  used  in  one  place  in  the  final  solution  it  is  superfluous  in  the  other. 

With  the  value  of  ct  determined  from  the  second  of  (45),  the  solution 
of  (43)  satisfying  the  periodicity  and  initial  conditions  is 


„      *>  AJQlsllsZ 1 

•^3  0/1     I     o  A    \  A          i 


2(1+  2A0)A0  '      2(1  -  7A0+isAl)A0 


~^==smVA0t ^-f;— Tjsm 

64(1-7^+18^)^0 


(46) 


where  c3  is  so  far  undetermined. 

The  equation  for  the  determination  of  z4  is 

z"<+A^  =  -A'z2+3BQ(xzz,+x3z1}  +  f  <70z*z2.  (47) 

In  order  that  the  solution  of  this  equation  shall  be  periodic  the  coefficient 
of  sin  V~A~0 1  in  its  right  member  must  equal  zero.  This  function  of  t  arises 
from  every  term  in  the  right  member  of  (47),  and  it  follows  from  (40),  (42), 
and  (46)  that  its  coefficient  carries  c2  linearly  and  homogeneously.  Therefore 
this  condition  determines  c2  uniquely  by  the  equation  c2  =  0,  whence 

z2=.T3=i/3=0.  (48) 

After  the  sign  of  c,  has  been  chosen  all  the  other  ct  are  determined 
uniquely  by  the  conditions  that  all  the  z<  separately  shall  be  periodic.  For, 
suppose  that  xlt  .  .  .  ,  x^;  ylt  .  .  .  ,  y^;  zlt  .  .  .  ,  zt^  have  been 
computed  and  that  their  coefficients  are  entirely  known  except  the  arbi- 
trary terms  c^-j/VZ^  sin \/3^  t  in  z,_2  and  c4_i/v^  sin  VA~0  in  zt^ ,  and 
the  arbitrary  constant  c,_2 ,  which  enters  linearly  in  x^  and  yt^  .  The  zt  is 
defined  by 

z?+A0zi=-A'zi_t+3B,(XtZt_t+xi_1z1)  +  ^C0zlz<_2+  •  •  •  ,        (49) 

where  the  terms  not  written  are  completely  known.  The  arbitrary  c4_2 
enters  linearly  in  the  coefficient  of  sin  v^  t  in  the  right  member  of  this 
equation,  which  does  not  involve  c(-1}  and  the  constant  ct^3  is  uniquely 
determined  by  the  condition  that  this  coefficient  shall  vanish. 


OSCILLATING   SATELLITES — SECOND   METHOD.  213 

It  can  !>»•  -hown  without  difficulty  thut  the  solutions  have  the  following 
propertii 

1.  The  xt,n,  ytj+t,  z^,  are  identically  zero  (j-1,  2,  ...«). 

2.  The  xlt  y,,  z,  involve  r,  homogeneously  to  the  degree  j. 

3.  The  xtl  are  sums  of  COHIM->  of  even  multiples  of  vT.J,  the  highest 

multiple  being  2j. 

4.  The  ytl  an   >ums  of  sines  of  even  multiples  of  VA~tt,  the  highest 

multiple  being  2j. 

5.  The  ztt+t  are  sums  of  sines  of  odd  multiples  of  \/~A~tt,  the  highest 

multiple  being  2j+l. 

(i.  Changing  the  sine  of  c,  is  equivalent  to  changing  the  sign  of  X', 
which  is  equivalent  to  increasing  t  by  r/^/~At.  Therefore  the 
two  values  off,  (or  X*)  belong  to  the  same  physical  orbit,  the 
origin  of  time  being  different  by  half  a  period  in  the  two  cases. 

7.  The  orbits  are  symmetrical  with  respect  to  the  x-axis,  the  xy-pl&ne, 
and  the  zz-plane. 

It  is  not  necessary  to  go  into  the  proofs  of  these  properties,  which  are  the 
same,  so  far  as  the  comparison  can  be  made,  as  those  found  in  §87. 

102.  Direct  Construction  of  the  Solutions  for  Class  B. — For  these 
orbits  it  is  advantageous  to  use  the  first  four  equations  of  (7),  the  last  one 
being  identically  zero.  We  have  proved  that  the  solutions  are  expansible 
as  power  series  in  X*,  that  the  coefficients  of  each  power  of  X*  are  periodic 
with  the  period  2T/a0 ,  and  that  z'  =  0  at  t  =  0  for  all  X. 

The  coefficients  of  X'  are  defined  by  the  differential  equations 


(50) 
Owiu  =  0. 

The  periodic  solutions  of  these  equations  are  seen  to  be 

From  equations  (6)  and  the  initial  value  of  x'  it  is  found  that 

/•\  (1)  •  —(I)    rt  4  "*)  • 

a,l  =  at=  —  a   ,  Xi  =  a     cos<r0t,          I/i=~w»Q     sm<r, I,         v.^*/ 

where  the  cot>fficient  a(l)  is  so  far  undetermined. 


214  PERIODIC    ORBITS. 

The  coefficients  of  X  are  defined  by 


.(2)  *ii  r     i(2)  \    \  <2> 

ti  •     (<>)  i  *''(\\ 

jr     411^'   I        "I  J _ 


dt  (m0ffo  —  n0po)  V  —  1 

(2) 

\ff  t-^(2)  _ 


3       _  ^(2)  _   ___     o  __  I 

dt  (mo  <r0—  no  PO) 


(2)  t        I  (2) 

I        f 


(m0o-0—  rioPo) 
where 


The  periodic  solutions  of  these  equations  satisfying  x'  =  Q  are 


a(2> 


where 

a<2)  is  so  far  undetermined, 


°'""    -)-Wopo)po 

?)  (a(1))2  »'„ 


<2) 


0—  rz0p0)o-0 


"o    <7o 

0  g0  a0  (aa))2 


<»  =   ,          00  0 

-  — 


(54) 


(oo; 


OSCILLATING    SATELLITES  —  SECOND   METHOD.  215 

The  arbitrary  a"'  is  determined.  except  as  to  sign,  by  the  periodicity 
condition  in  the  next  step  of  the  integration;  and  the  double  sign  is 
equivalent  to  the  double  sign  on  X*.  After  «'"  ha>  been  determined,  an 
a"1  is  uniquely  determined  by  the  periodicity  condition  at  each  succeeding 
step  of  the  integration. 

The  coefficients  of  X1  are  defined  by 

_  r  tV»  -  .     *j 


dt 

0       *    "  i  HI.  a.—  II.  n.\   v'_  1  (m.  a-4-n.a.} 

(57) 


._,  rtv- 

dt 


dt 
where 

[    ]U)=-f 


_ 

••  (m0<r.-n,p,)  (m.  p.+n.  < 

u«_  ,       >u  i"  " 

M4'  -  h 


(58) 


In  order  that  the  solution  of  the  first  equation  shall  be  periodic  it  is 
necessary  that  in  its  right  member  the  coefficient  of  e"^7'  be  equal  to  zero. 
That  part  of  this  coefficient  which  arises  from  A'z,  involves  a(1)  linearly  and 
homogeneously.  Those  parts  which  arise  from  x,  yt ,  xj  etc.  carry  (a(l>)'  as 
a  factor  and  involve  it  in  no  other  way.  Consequently,  the  condition  that 
the  coefficient  of  e*tvrrr'  shall  vanish  is  satisfied  by  a(l)  =  0,  or  by  an  equa- 
tion of  the  form 

P(a(U)'+Q  =  0,  (59) 

where  P  and  Q  are  known  constants.  It  is  easily  shown  that  they  are 
identical  with  the  coefficients  of  oj  and  aX  which  arise,  in  §100,  in  the 
demonstration  of  the  existence  of  the  solutions.  The  first  determination 
of  at  leads  to  the  trivial  solution  x=y=0;  equation  (59)  gives  the  double 
determination  for  a<l}  mentioned  on  page  210. 

In  order  that  the  solution  of  the  second  equation  shall  be  periodic  it 
is  necessary  that  in  its  right  member  the  coefficient  of  e~"v=T<  be  equal 
to  zero.  It  is  easy  to  see  that  this  condition  determines  a(l>  by  an  equation 
which  is  identical  with  (59).  That  is,  the  same  value  of  a'"  makes  the 
solutions  of  both  the  first  and  the  second  equations  periodic. 

The  particular  integrals  of  the  third  and  fourth  equations  are  periodic. 
In  solving  the  third  and  fourth  equations  the  constants  of  integration  are 
always  to  be  taken  equal  to  zero. 


216  PERIODIC    ORBITS. 

The  right  members  of  the  differential  equations  for  the  terms  of  the 
next  order  are 


(60) 


Before  integrating  the  first  equation  the  coefficient  of  ean/^'  in  its  right 
member  must  be  put  equal  to  zero.  It  is  found  from  an  examination  of 
the  terms  of  (60)  that  a(2)  is  involved  linearly  but  not  homogeneously. 
Moreover,  a(2)  is  the  only  unknown  quantity  in  this  coefficient.  Therefore 
a(2)  is  uniquely  determined  by  setting  the  coefficient  of  $""•'=>'  equal  to  zero. 
The  condition  that  the  solution  of  the  second  equation  shall  be  periodic 
is  identical  with  that  imposed  by  the  first.  The  particular  integrals  of  the 
third  and  fourth  equations  are  periodic. 

At  the  ilh  step  the  right  members  of  the  differential  equations  involve 


(61) 


where  the  parts  not  explicitly  written  are  independent  of  a"  2).  In  order 
that  the  solution  of  the  first  equation  shall  be  periodic  it  is  necessary  that 
the  coefficient  of  eff°v=It  in  its  right  member  be  put  equal  to  zero.  This 
coefficient  carries  a(<~2)  linearly  and  in  general  non-homogeneously,  and  the 
known  factor  by  which  a(<~2)  is  multiplied  is  precisely  the  same  as  that  of 
aw  in  the  equation  by  which  the  latter  was  determined.  Therefore  the  arbi- 
trary a(i~2)  is  uniquely  determined  by  setting  the  coefficient  of  e'"v=li  equal 
to  zero,  for  it  carries  no  other  unknown.  When  this  condition  is  satisfied 
the  coefficient  of  e-ffov=Tl  in  the  second  equation  is  zero,  and  the  entire  solu- 
tion at  this  step  is  periodic.  Therefore  after  the  sign  of  a(1)  has  been  chosen 
the  process  is  unique,  and  it  can  be  continued  indefinitely. 


CHAPTER  VII. 

OSCILLATING  SATELLITES  WHEN  THE  FINITE 
MASSES  DESCRIBE  ELLIPTICAL  ORBITS. 

103.  The  Differential  Equations  of  Motion.  Suppose  the  finite  bodies 
describe  ellipses  whose  eccentricity  is  c.  Let  1  —  n  and  n  (/j^0.5)  represent 
their  masses,  and  then  determine  the  linear  and  time  units  so  that  their  mean 
distance  apart  and  the  gravitational  constant  shall  be  unity.  With  these 
units  their  mean  angular  motion  is  unity. 

Now  refer  the  system  to  a  set  of  rectangular  axes  with  the  origin  at 
the  center  of  gravity,  and  let  the  direction  of  the  axes  be  so  chosen  that  the 
£7>-plane  is  the  plane  of  motion  of  the  finite  bodies.  Suppose  the  £i?-axes 
rotate  with  the  constant  angular  rate  unity  around  the  f-axis  in  the  direction 
of  motion  of  the  finite  masses,  and  suppose  1— M  and  /*  are  on  the  {-axis 
when  they  are  at  the  apses  of  their  orbits.  Then  the  differential  equations 
of  motion  for  the  infinitesimal  body  are 


(1) 


<#  (1— M)(n-ih) 

dt  i\  rj 


where 


and  where  £,  ,  ^,,  77,  ,  and  ijt  are  determined  by  the  fact  that  the  finite  bodies 
move  in  ellipses. 

If  we  let  p,  and  t>,  be  the  polar  coordinates  of  1  —  n  referred  to  fixed 
axes  having  their  origin  at  the  center  of  mass  of  the  system,  and  p,  and  t'f 
the  corresponding  coordinates  of  /*»  we  have 


,  =  -p,  cos(»,-0,  &  = 

J,=  -p.sin^.-O,  77,= 


|l-ecos<+  j  (1  -cos  20+  •  •  •}» 


(2) 


where  the  initial  value  of  t  has  been  so  determined  that  the  bodies  are  at 
their  nearest  apses  at  (  =  0.  n7 


218  PERIODIC    ORBITS. 

104.  The  Elliptical  Solution. — In  order  to  get  the  Lagrangian  elliptical 
solution  for  the  infinitesimal  body  we  consider  first  the  two-body  problem. 
The  equations  of  motion  for  the  infinitesimal  body  subject  to  the  attraction 
of  a  mass  m  are,  when  referred  to  the  rotating  axes, 


=~k*m> 


(3) 


We  shall  consider  the  solution  in  the  ^-plane.  If  the  eccentricity  of  the 
orbit  is  e  and  if  the  mean  motion  with  respect  to  the  fixed  axes  is  unity, 
then  the  solution  with  the  same  determination  of  the  origin  of  time  and 
apses  as  in  (2)  is 


(4) 


v  —  f),          r?  =  rsin(v  —  f), 

(l-cos20+ 


It  will  now  be  shown  that  equations  (1)  will  be  satisfied  if  the  infini- 
tesimal body  moves  so  that  the  ratios  of  its  coordinates  to  the  corresponding 
coordinates  of  the  finite  masses  have  certain  constant  values.  Let  the 
coordinates  in  these  special  solutions  be  represented  by  £0  ,  rj0  ,  and  0;  then 


t    -  i  _       2,  ,rx 


Upon  making  use  of  (2)  and  (5),  it  is  found  that 

t 

i»— %™  + 


(6) 


SATKI.UTKS,  KI.UPTICAL  r\  219 

Then  equation-     1  i  heroine 


r* 


., 

ft 


(7) 


The  first  two  of  these  equations  are  of  the  same  form  as  (3),  and  their  solu- 
tions eorresjMmding  to  (4)  are 


From  these  equations  and  (2),  we  find 
&  _.  30  _       [(l 

«l  Til 

and  from  (6), 


On  equating  these  two  expressions  for  the  ratio  £,/£,  =  JJ./T;,  ,  and  rationalizing, 
we  have 

M  =  0.  (9) 


It  is  easily  verified  that  starting  from  the  expressions  for  the  ratio 
the  same  quintic  equation  is  obtained.  Therefore,  for  those  values  of  M 
satisfying  (9),  equations  (2)  and  (8)  are  a  particular  solution  of  the  thn-e- 
hody  problem  where  one  mass  is  infinitesimal.  As  is  well  known,  there 
:ire  three  real  solutions,  one  for  each  ordering  of  the  three  masses.  As  in 
Chapter  V,  we  shall  call  them  (a),  (6),  and  (c)  in  the  order  of  decreasing 
values  of  their  x-coordinates. 

Equation  (9)  is  Lagrange's  quintic  in  case  one  mass  is  infinitesimal  and 
the  units  are  chosen  so  that  the  masses  of  the  finite  bodies  are  1-p  and  ». 
For  example,  if  in  equation  (60),  page  216,  of  Introduction  to  Crlextial 
s.  we  put  /»,  =  !-//,  »i,=0,  and  W,=M,  we  get  equation  (9). 


220  PERIODIC    ORBITS. 

105.  Equations  for  the  Oscillations.  —  We  shall  study  the  oscillations  in 
the  vicinity  of  the  Lagrangian  solutions.  For  this  purpose  we  make  the 
transformation 

(10) 


in  equations  (1),  and  expand  as  power  series  in  x,  y,  and  z.     After  this 
transformation  and  expansion,  we  let 


in  those  places  where  /z  appears  explicitly.  This  is  not  the  only  way  in 
which  the  original  p  can  be  divided  into  the  new  ju  and  ju0+^>  and  sometimes 
others  are  advisable.  The  coefficients  of  the  various  powers  of  x,  y,  and  z 
are  expansible  as  power  series  in  e,  the  terms  independent  of  e  being  constants, 
as  is  seen  from  (2)  and  (8).  We  find  from  (6)  and  (8)  that 


+2esinH-      si 


= 


(12) 


Consequently,  after  making  use  of  these  expansions  and  the  transformations 
(10)  and  (11),  equations  (1)  become 


4 


x 


1* 


(13) 


•  »(  ll.I.AIIN.,    SATBLUTBB,    Kl.I.ll'TICAL    C.\  •_'_'! 

where 

I  -     L  '"Mo,     Mo 
•• '  —  "1       _<o)i     T  _(o)»  ' 


A"  =  +  \  -  --S5i  +  '(»  ,  -<>    ,.<o)j  -  L   re.-  /  .- 

I       'i  't  L'I  r»    J 

y\ 


rJ»*  -  r»»J  fcos<+   •  •  •     y\ 


ccost+ 


-\z\ 


(14) 


tho  signs  in  the  [  ]  being  -f  +  ,  +  -  ,  •  -  ,  according  as  the  jx)int  (a), 
(6),  or  (c)  is  under  consideration.  All  terms  up  to  the  second  order  inclusive 
in  x,  y,  z,  and  X  are  written. 


222  PERIODIC    ORBITS. 

106.  The  Symmetry  Theorem.  —  It  follows  from  (2),  (8),  and  (1)  that 
the  right  members  of  (12)  have  the  following  properties: 

(a)  The  X  and  Y  involve  only  even  powers  of  z,  and  Z  involves  only 
odd  powers  of  z. 

(b)  In  X  the  coefficients  of  all  terms  involving  even  powers  of  y  are 
sums  of  cosines  of  integral  multiples  of  t,  and  the  coefficients  of 
all  terms  involving  odd  powers  of  y  are  sums  of  sines  of  integral 
multiples  of  t. 

(c)  In  Y  the  coefficients  of  all  terms  involving  even  powers  of  y  are 
sums  of  sines  of  integral  multiples  of  t,  and  the  coefficients  of  all 
terms  involving  odd  powers  of  y  are  sums  of  cosines  of  integral 
multiples  of  t. 

(d)  In  Z  the  coefficients  of  all  terms  involving  even  powers  of  y  are 
sums  of  cosines  of  integral  multiples  of  t,  and  the  coefficients  of 
all  terms  involving  odd  powers  of  y  are  sums  of  sines  of  integral 
multiples  of  t. 

Suppose  the  initial  conditions  are 

x  =  aj         x'  =  o,         y  =  Q>         y'  =  ft,         z  =  0,          z'  =  y.          (15) 

Then  the  solutions  of  (12)  are 

x=f(a,  0,  0,  ft,  0,  7;  0,  x'=f(a,  0,  0,  ft,  0,  7;  0, 

y  =  g(a,  0,  0,  ft,  0,  7;  0,  y'  =  &'(<>•,  0,  0,  0,  0,  y;  t),  (16) 

z=h(a,  0,  0,  0,  0,  7;  0,  «'  =  /i'(a,  0,  0,  ft,  0,  y;  t). 

Now  make  the  transformation 
x=+x,    x'=-x',    y=-y,     y'=+y',    z=-z,     z'=+z',    t=-t.  (17) 

It  follows  from  the  properties  (a)  ,  .  .  .  ,  (d)  that  the  form  of  equations 
(13)  is  not  changed  by  this  transformation.  Consequently  the  solutions 
with  the  initial  conditions 

x  =  a,          x'  =  0,          y  =  0,          y'  =  ft,          «  =  0,          z'  =  y, 
are  identical  with  (16),  and  we  have,  making  use  of  (17), 


(18) 


Therefore,  with  the  initial  conditions  (15),  x,  y',  and  z'  are  even  functions  of 
t,  while  x',  y,  and  z  are  odd  functions  of  t.  That  is,  if  the  infinitesimal 
body  crosses  the  z-axis  perpendicularly  when  the  finite  bodies  are  at  an 
apse,  its  motion  is  symmetrical  with  respect  to  the  it-axis. 


u.-<  II.I.AIIM,  >\  i  u.u  n..-.  Ki.i.ii'in  \i.  i  \-i.  •_'_':; 

107.  Integration  of  Equations  (13).—  The  Terms  of  lh<   f-'irst  It, ,,,,,. 
Suppose  tlu-  initial  conditions  are 

' <  =  a,.    *'(())  =  a,,    j/(0)  =  alt    J/'(0)  =  a4)    z(0)  =  a»,    z'(0)  =  a,.     (19) 

\\  e  shall  now  integrate  c(|iiations  (13)  as  power  series  in  a,,  a,,  a,,  a4  , 
a>  ,  at  ,  and  X.  Since  there  are  no  terms  in  the  differential  e<|iiation>  inde- 
pendent  of  .r.  //,  and  2  and  their  derivatives,  there  will  he  no  terms  in  the 

solutions  independent  of  a, a,. 

The  terms  of  the  first  degree  in  a ,  a,  are  defined  liy  the  differ- 
ential equations 


20) 


-,-  [l-.l  - 
z'+  [A  +3 Accost  +  f  4e*(l+3cos20  +  •  •  •  ]  z,  =0, 


subject  to  the  initial  conditions  (19).     The  coefficients  are  power  series  in 
<•.  periodic  with  the  period  2ir,  and  reduce  to  constants  for  r  =  0.     The  first 
two  equations  are  independent  of  the  third,  and  conversely. 
For  e  =  0  equations  (20)  become  simply 

x'l-2y[-(].+2A}xl=Q,         yi+2xfl-[l-A]yl=0,         z't+Azt=Q.     (21) 


From  the  results  obtained  in  §§23-25  it  follows  that  the  properties  of  the 
solutions  of  (20)  depend  upon  the  character  of  the  roots  of  the  characteristic 
equation  of  (21).  This  equation  for  the  first  two  of  (21)  is 

«4+(2-^)s-*+(l-yl)(H-2^)=0.  (22) 

Two  roots  of  this  equation  can  be  equal  only  if  ^4  has  one  of  the  values 
-  1/2,  0,  S/9.  or  1.  The  first  two  are  excluded  by  the  fact  that  A  is  necessarily 
po>itive.  When  M,  is  near  n  in  value,  as  it  will  always  be  taken  here,  A  is 
greater  than  unity  and,  consequently,  the  last  two  values  are  excluded. 

Equation  (22)  has  two  pairs  of  roots  equal  in  numerical  value  but 
opposite  in  sign,  and  for  A>\  two  of  them  are  pure  imaginaries  and  two 
are  real.  Let  us  represent  them  by  ±<rtV—  I,  ±p0,  where  <TO  and  p0  are 
real.  We  now  raise  the  question  whether  two  of  the  roots  of  (22)  can  differ 
by  an  imaginary  integer.  In  order  that  this  may  be  so  we  must  have 


where./  is  an  integer.     Since  this  value  of  «  must  satisfy  (22),  we  find 

j4  —  4(2—A)jt+l6(l—A)(l  +  2A)  =  Q; 

whence  

f!44.  (23) 


224  PERIODIC    ORBITS. 

The  question  is  whether  there  are  integral  values  of  j  giving  admissible 
values  for  A  by  (23) .  To  insure  the  convergence  of  the  final  series  it  will  be 
necessary  to  take  MO  nearly  equal  to  n,  and  we  shall  suppose  at  once  that 
this  condition  is  satisfied.  It  was  shown  in  §82  that  when  /z0  =  /x  the  value 
of  A  exceeds  unity  for  each  of  the  points  (a),  (b),  and  (c)  for  all  values  of  /*• 
From  the  expansions  given  in  equations  (42),  p.  206,  of  Introduction  to 
Celestial  Mechanics,  it  is  seen  that  for  very  small  values  of  n  the  values 
of  A  are  approximately  4,  4,  and  1  for  the  points  of  libration  (a),  (b), 
and  (c)  respectively.  In  the  numerical  example  of  §90,  for  which  ^  =  1/11, 
it  was  found  that  A  is  2.25,  6,51.  and  1.08  for  the  points  of  libration  (a), 
(6),  and  (c)  respectively.  In  the  extreme  case  of  /*—  1/2  we  easily  find  from 
the  formulas  of  §§76  and  82  that  the  values  of  A  are  1.56,  8,  and  1.56  for 
the  points  of  libration  (a),  (b),  and  (c)  respectively.  While  fj.  varies  from 
0  to  1/2  the  values  of  A  vary  from  4  to  1.56,  4  to  8,  and  1  to  1.56  for  the 
points  (a),  (b),  and  (c)  respectively.  From  this  we  see  what  values  of  A  are 
possible  in  the  actual  problem,  and  we  shall  be  able  to  determine  whether  j 
takes  integral  values  for  any  of  them. 

When  the  negative  sign  is  taken  before  the  radical  in  (23),  the  function 
16  .A  has  its  greatest  value  of  zero  for  J2  =  4.  Consequently  we  get  no 
admissible  values  of  A  using  this  sign  before  the  radical. 

When  the  positive  sign  is  taken  before  the  radical  in  (23),  j  and  A  are 
found  to  have  the  following  relations: 

If      j  =    1    ,        2    ,        3   ,        4   ,        5    ,       6    ,   .  .  .  , 
then  A  =0.92,      1.00,     2.01,     3.73,      5.94,      8.70,..., 

and  A  is  larger  than  8.7  for  all  j  larger  than  6.  The  values  of  A  for  j 
between  2  and  6  are  admissible  for  either  the  point  (a)  or  the  point  (b), 
but  not  for  the  point  (c).  Consequently  since  A,  and  therefore  tr,  is  a 
continuous  function  of  ju  and  /*„ ,  there  exist  pairs  of  values  of  M  and  n0  for 
which  the  difference  of  the  imaginary  roots  of  (22)  is  an  imaginary  integer, 
but  they  are  exceptional. 

The  characteristic  equation  for  the  third  equation  of  (21)  is  simply 


Hence  the  solution  of  the  third  equation  of  (20)  does  not  take  an  exceptional 
form  unless  2  VZ  is  an  integer,  or 

72 

A  =  *j-  (j  an  integer). 

It  follows  from  the  discussion  above  that  the  admissible  values  of  A  satis- 
fying this  relation  are  2.25,  4.00,  and  6.25.  These  values  belong  only  to 
the  point  (a)  or  the  point  (b),  and  in  no  case  can  the  congruence  be  satisfied 
for  the  point  (c). 


•  >M  II.I.MIM.    -\  I  F.I.I.  II'KS,  ELLIPTICAL   CASE.  '2'2~i 

According  to  tin-  results  obtained  in  §2:i,  the  solutions  and  character- 
istic exponent.-  of  I'D  ;ire  always  expansible  as  power  series  in  »  except  when 
.1  has  the  special  values  noted  above:  and  according  to  the  results  obtained 
in  §24,  the  same  result  is  true,  in  general,  even  if  the  roots  of  the  character- 
i.-tic  e(iuations  differ  by  imaginary  integers.  However,  in  the  latter  case 
the  construction  of  the  solutions  is  quite  different. 

It  was  proved  in  §M  that  in  equations  of  the  type  under  consideration 
here  the  characteristic  exponents  occur  in  pairs  which  are  equal  numerically 
luit  opposite  in  sign.  Therefore1  the  solutions  of  (20)  arc  of  the  form 


+atepl[pul+u't]+a<e-»[-put+u't}, 


where 

«,  ,  .  .  .  ,  at,  ct  ,  Cf,  arc  arbitrary  constants  of  integration  , 


(24) 


P  =     p.+p,e+p1e!-r- 


>    (25) 


Mi0>>  "«w>  u'«<0)  &re  constants. 

The  initial  values  of  the  v,  and  w,  can  be  taken  equal  to  unity  without 
loss  of  generality,  and  will  be  so  chosen.  Moreover,  the  t/,.  vt  ,  and  u\  are 
periodic  with  the  period  2*-,  and  since  this  property  holds  for  all  e  for  which 
the  series  converge,  each  ?/{",  v\'\  and  w\"  separately  is  j>criodic  with  the 
period  2r.  The  coefficients  of  these  series  can  be  found  by  the  methods 
set  forth  in  §26.  The  a,  and  r,  are  uniquely  expressible  in  terms  of  the 
initial  values  of  x,  x',  y,  y',  z,  and  z',  the  a,  of  equations  (19),  because 
the  solutions  (24)  constitute  a  fundamental  set,  by  hypothesis,  and  the 
determinant  of  the  coefficients  of  the  a,  and  r,  is  therefore  distinct  from 
zero.  We  may  use  either  the  a,  and  c,  or  the  a,  as  arbitrarics. 

The  characteristic  exponents  <r,  p,  and  u  are  real  for  e  sufficiently 
small,  as  we  shall  now  show.  The  w  arises  from  the  third  equation  of  (21), 
which  is  of  the  same  form  as  that  treated  in  §50,  where  it  was  shown  that 
the  characteristic  exponents  are  pure  imaginaries  in  this  case.  The 
±0V—  1  and  ±p  are  roots  of  an  equation  of  the  form 

A(a,  e)-0,  (26) 


226  PERIODIC   ORBITS. 

where  A  is  an  even  function  of  a  and  a  power  series  in  e  [see  §22,  and  in 
particular  equation  (98)].     For  e  =  0  the  solutions  of  this  equation  are 


where  o-0  and  p0  are  real.     Now  let 

a  =  o-0  V 
and  (26)  becomes 

A(<r0V=T,  e)+A'(<r0V^T,  e)/3V^7  -  lA"(<r0v^T,  e)^+  .  .  .  =0.    (27) 

All  the  even  derivatives  are  even  functions  of  <r0  V  —  1  ,  and  the  coefficients 
of  the  various  powers  of  0  in  these  terms  are  therefore  real;  all  the  odd 
derivatives  are  odd  in  cr0V  —  1  and  are  multiplied  by  odd  powers  of 
/3V—  l,  and  the  coefficients  of  the  various  powers  of  /3  in  these  terms  are 
therefore  also  real.  Consequently,  since  A'  (<70  V—  l,  0)  is  distinct  from 
zero  under  the  conditions  satisfied  in  this  problem,  the  solution  of  (27)  for 
|8  as  a  power  series  in  e,  vanishing  with  e,  gives  a  unique  series  whose  coeffi- 
cients are  ah1  real.  Therefore  a  =  a  is  real.  It  can  be  shown  similarly  that 
p  is  a  real  constant. 

Now  suppose  the  initial  conditions  are  given  by  (15).     Then,  since 

yi(0  =  -|/i 
we  have 


eus/=T(M;2  (  — 

On  applying  the  lemma  of  §58,  with  the  necessary  slight  modifications,  it 
follows  that 

al=-a2,        a}=-at,        ^=-0^. 

Then  we  have  from  (24) 


XC  -  0]  +  «.,  [e-p(w3(  -0  ~  e+p(  M4(  -  «)]  , 


ILLATlNi;    SATELLITES,    ELLIPTICAL   CASE. 

Since  tin-!-  relation*  arc  idrntitio  in  t.  we  have 


227 


Therefore,  if  the  tfc,  r  .  and  »•  arc  arranged  as  Fourier  scries,  they  satisfy 
the  relations 


•M,  =  +2  [A,  cosjt+B,  xinjt], 
«,=  -2  [.-I/  cosjl-Bj  smjt], 
M,  =  +2  [C;  cosjt+D;  sin  j<], 
M«  =  -  2  [C,  cosjf  -  Dy  sinj/], 
M-,  =  +2  [Kj  cosjt+Lj  sinjt], 


f,  =  +2  [£,  cosjt+F,  smjt], 
r,  =  +2  [£,  cosjt-Fj  smjt], 
v,  =  +2  [G,  cosjt+H,  smjt], 
vt  =  +2  [G/  cosjf  -  //,  sinjf], 
tr,=  +  2  [K,  cosjt-L,  sinjt]. 


(38) 


Since  the  i/;,  v},  and  uv  are  defined  by  linear  equations  they  are  indepen- 
dent of  the  initial  conditions.     But  the  equations  a,=  -a,,  a,--a«,  c,=  -c, 

liold  only  for  the  initial  conditions  (5). 

108.  The  terms  of  the  Second  Degree.  —  The  terras  of  the  second  degree 
in  the  a,  and  X  are  found  from  equations  (13)  and  (14)  to  be  defined  by 


(29) 


, 


x't-2y't-[l+2A+6Aecost+ 
y'+2xt-[BAe8\nt+  •  •  •]  x,  -  [l-A-3Aecost+ 
g  +  [  A  +  ZAecost+    •  •]  2,  =  Zj 

\\liere 
Xt=+[-2A'-  6 

-3B  --  l2Becost-\ 
B  +    GBecost+ 


--  6A'esmt+  •    -]yt\ 


A'+  3A'ecost+  •  • 
[-\2Bes\nt+---]x\  +[3B  +  l2Becosl+  •  •  •]*,!/, 
9Bc  8inl+---!/I  +          +    3Besint+  •  • 


6Bes\nt-\ 


(30) 


228  PERIODIC   ORBITS. 

It  is  necessary  for  further  work  to  determine  some  of  the  properties  of 
the  solutions  of  equations  (29).  These  properties  depend  upon  the  form 
and  properties  of  their  right  members,  which  are  given  in  equations  (30). 
The  general  character  of  the  coefficients  in  these  equations  easily  follows 
from  the  properties  (a),  .  .  .  ,  (d)  of  §106.  On  referring  to  the  results  which 
were  developed  in  Chapter  I,  it  is  found  that: 

[1]  The  solutions  of  (29)  consist  in  the  first  place  of  the  complementary 
functions,  and  they  are  identical  in  form  with  (24).  The  arbitraries 
a,  and  c,  which  appear  are  uniquely  determined  by  the  conditions 
x,  (0)  =  x(  (0)  =  yt  (0)  =  y'a  (0)  =  z2  (0)  =  z\  (0)  =  0,  where  z2  ,  .  .  .  ,  z'2  are 
the  complete  solutions  of  (29)  [§15]. 

[2]  There  are  terms  arising  from  those  parts  of  the  right  members  which 
contain  X  as  a  factor.  These  right  members  consist  of  sums  of  terms 
which  are  periodic  with  the  period  2ir  multiplied  by  one  of  the  funda- 
mental exponentials  e+<rV^',  e-ffV^' ,  e+pt ,  e~pt ,  ea^' ,  and  e-^'~< . 
Hence  it  follows  from  the  results  of  §30  that  the  corresponding  parts 
of  the  solutions  consist  of  sums  of  periodic  terms  whose  period  is  2ir 
multiplied  by  these  same  exponentials,  plus  t  times  the  corresponding 
part  of  the  complementary  function.  We  shall  be  particularly  interested 
in  those  terms  which  contain  t  as  a  factor.  They  are  linear  in  the  a( 
and  c, ,  and  for  e  =  0  the  parts  having  the  period  2  ?r  reduce  to  con- 
stants. The  expressions  for  x2  and  yt  do  not  depend  upon  ct  and  c2 , 
and  the  expression  for  z2  is  independent  of  at ,  .  .  .  ,  a4 . 

[3]  Next  consider  the  parts  of  the  right  members  of  (29)  which  are  inde- 
pendent of  X  .  They  consist  of  sums  of  periodic  terms  having  the  period 
27r  multiplied  by  the  squares  and  second-degree  products  of  the  funda- 
mental exponentials.  Therefore,  except  in  the  special  case  where 
a  or  co  is  an  integer,  the  exponents  of  the  exponentials  in  the  right 
members  are  not  congruent  to  any  of  the  characteristic  exponents 
mod.v^T;  hence  it  follows  by  §30  that  corresponding  parts  of  the 
solutions  consist  of  sums  of  periodic  terms,  period  2?r,  multiplied  by 
these  same  exponentials.  In  particular,  there  are  no  terms  containing 
t  as  a  factor.  These  terms  are  homogeneous  of  the  second  degree  in 
the  a,  and  the  c, . 

109.  The  Terms  of  the  Third  Degree. — The  terms  of  the  third  degree 
in  the  af ,  ct,  and  X  are  defined  by  equations  whose  left  members  are  identical 
in  form  with  the  left  members  of  (29).  The  right  members  contain  terms 
which  are 

(a)  linear  in  X  and  of  the  second  degree  in  x1 ,  yl}  and  z, ; 

(b)  linear  in  X  and  of  the  first  degree  in  xt,  yt,  and  z2 ; 

(c)  of  the  third  degree  in  xlt  yl}  and  z^  and 

(d)  of  the  first  degree  in  xu  ylt  zl  and  in  x2,ys,  and  z,. 


OSCILLATINC,    SATELLITES,   ELLIPTICAL   CASE.  229 

The  solutions  have  the  following  properties: 

[4]  There  an  the  complementary  functions  identical  in  form  with  the 
expressions  (24).  The  constants  which  appear  in  them  are  determined 
by  the  conditions  that  *,(0)  =*;(()) -y,(0)-y;(0)=z,(0)-«i(0)=0. 

(oj  The  part  of  the  solution  coming  from  the  terms  (a)  consists  of  sums 
of  periodic  terms,  period  2r,  multiplied  by  the  squares  and  second- 
dejiree  products  of  the  fundamental  exponentials. 

[0]  Tin-  terms  (b)  give  rise  to  terms  of  two  different  cla.sses  because  a^, 
yt,  and  z,  consist  of  terms  of  two  different  tyj>es.  There  are  terms 
which  contain  X  as  a  factor,  and  a  part  of  these  are  sums  of  periodic 
terms,  period  2*-,  multiplied  by  t  times  the  fundamental  exponentials; 
the  remaining  part  lacks  the  factor  t.  These  terms  are  homogeneous 
and  linear  in  the  (it  and  the  c, .  The  corresponding  parts  of  the  solu- 
tions are  sums  of  periodic  terms,  period  2*-,  multiplied  partly  by  ? , 
partly  by  t,  and  partly  by  f.  They  are  all  homogeneous  and  linear 
in  the  a,  and  the  c,,  the  x  and  t/-terms  being  independent  of  the  ct,  and 
the  z-terms  of  the  a, . 

The  terms  of  the  solutions  coming  from  the  other  part  of  (b)  are 
sums  of  periodic  terms,  period  2r,  multiplied  by  squares  and  second- 
degree  products  of  the  fundamental  exponentials.  They  are  homo- 
geneous of  the  second  degree  in  the  a,  and  the  c, . 

[7]  The  terms  of  the  type  (c)  are  homogeneous  of  the  third  degree  in  the 
a,  and  the  c( .  They  consist  of  terms  of  two  classes,  the  first  of  which 
are  sums  of  periodic  terms,  period  2r,  multiplied  by  the  fundamental 
exponentials  to  the  first  degree;  and  the  second  of  which  are  sums  of 
periodic  terms,  period  2r,  multiplied  by  cubes  and  non-canceling  third- 
degree  products  of  the  fundamental  exponentials.  The  corresponding 
parts  of  the  solutions  consist  respectively  of  t  times  sums  of  periodic 
terms,  period  2r,  multiplied  by  the  fundamental  exponentials  to  the 
first  degree,  and  sums  of  periodic  terms,  period  2*-,  multiplied  by 
cubes  and  non-canceling  third-degree  products  of  the  fundamental 
exponentials. 

[8]  The  part  of  the  solution  coming  from  the  terms  (d)  consists  of  terms 
of  two  kinds,  the  first  depending  upon  those  parts  of  xt,  yt ,  and  z, 
which  contain  X  as  a  factor,  and  the  second  depending  upon  those  parts 
of  x,,  yt,  and  z,  which  are  independent  of  X.  The  parts  of  the  solu- 
tions corresponding  to  the  first  are  homogeneous  of  the  second  degree 
in  the  a,  and  the  c, ,  and  they  are  sums  of  periodic  terms,  period  2r, 
multiplied  by  squares  and  second-degree  products  of  the  fundamental 
exponentials,  and  some  of  these  products  contain  t  as  a  factor  while 
others  do  not.  The  other  parts  of  the  solutions  coming  from  the  terms 
(d)  have  the  proj>erties  of  those  coming  from  (c). 


230  PERIODIC    ORBITS. 

110.  General  Properties  of  the  Solutions.  —  It  will  be  necessary  to  use 
the  following  general  properties  of  the  solutions  : 

[9]  Since  the  right  members  of  the  first  two  equations  of  (13)  involve  only 
even  powers  of  z,  it  follows  that  x  and  y  are  even  functions  of  Cj  and  c2 
taken  together. 

[10]  Since  the  right  member  of  the  third  equation  of  (13)  is  an  odd  function 
of  z,  it  follows  that  z  is  an  odd  function  of  CL  and  c,  taken  together,  and 
that  z  identically  vanishes  for  d  =  ci  =  0. 

[11]  Since  the  right  members  of  the  first  two  equations  of  (13)  vanish 
identically  for  x  =  y  =  z  =  0,  but  not  f  or  x  =  y  =  0,  it  follows  that  x  and  y 
vanish  identically  for  at  =  •  •  •  —  a4  =  ct  =  c2  =0,  but  not  for 


[12]  Since  the  equations  reduce  to  those  having  constant  terms  for  e  =  Q, 
it  follows  that  the  sums  of  the  periodic  terms,  period  2ir,  reduce  to 
constants  for  e  =  0. 

111.  Conditions  for  the  Existence  of  Symmetrical  Periodic  Orbits.  —  The 

differential  equations  are  periodic  in  t  with  the  period  2ir.  Consequently 
the  period  of  the  periodic  solutions,  if  they  exist,  will  be  T  =  2nir,  where  n 
is  an  integer.  When  the  initial  conditions  are  such  that  the  orbit  of  the 
infinitesimal  body  is  symmetrical,  as  defined  in  §106,  then  sufficient  con- 
ditions for  the  existence  of  the  periodic  solutions  are 


(31) 


These  equations  are  power  series  in  a,,  .  .  .  ,  a4,  clt  c2,  and  X,  and 
vanish  identically  with  at  =  •  •  •  =  a4  =  c:  =  c2  =  0.  In  order  that  they  may 
have  any  solution  for  a:  ,  .  .  .  ,  at,  clt  and  c2  ,  vanishing  with  X,  aside  from 
this  one,  the  determinant  of  the  coefficients  of  the  linear  terms  in  a,  ,  .  .  .  ,  a4  , 
cu  and  c2  must  vanish.  It  follows  from  (28)  that 

«,«»—  U.CO), 


-»(0)  =0,  z         -2(0)  = 


(32) 


OSCILLATING    SATELLITES,  ELLIPTICAL   CASE. 


231 


and  therefore  the  determinant  of  the  coefficients  of  the  linear  terras  of  (31)  is 
found  from  (24)  to  be 

A-A.A,,  (33) 


1 

o 

rvcn± 
Ate          -Rl  ,    A, 


0 


0 
-»vrrr 


0 

1 


,(34) 


where 


—U  V  —  |  X 


(35) 


On  reducing,  we  find 

r  ,vrri  I        _r^ 
A,=  [e       7-e 

A,=  w,(-|)[euv        -e"" 

It  will  now  be  shown  that  t0,(7Y2)  and 

A,  At 


A, 


(36) 


are  distinct  from  zero.  Let  D,  represent  the  determinant  of  the  fundamental 
set  of  solutions  of  the  x  and  {/-equations,  and  Dt  that  of  the  z-eq nation. 
On  writing  these  determinants  for  the  time  t  =  T/2  and  making  use  of  the 
relations  (32),  we  find 


".(I). 


r 

'-'I     » 


*(f) 

-c, 


232 

where 


PERIODIC    ORBITS. 


The  determinants  become,  after  some  reductions, 


«.(!). 


X 


/T\ 
V*\2)  ' 


(37) 


Since  A  and  D2  are  determinants  of  fundamental  sets  of  solutions  of 
linear  differential  equations  for  the  regular  point  t  =  T/2,  they  are  distinct 
from  zero.  Therefore  their  second  factors  are  not  zero,  and  equations 
(36)  can  be  satisfied  only  by 


r 
\ 


—  e 


:*]=0  or  [euV-l*-e-uV-**]=0. 


(38) 


If  either  of  these  equations  is  satisfied,  A  is  zero. 

In  order  that  one  of  equations  (38)  shall  be  satisfied  it  is  necessary 
that  either 

aT  =  2N1w,  or  wT  =  2NsTr  (N,,  N,  integers).  (39) 

Since  T  =  2mr,  where  n  is  an  integer,  these  conditions  become 


a  =  — -  ,  or  co  = 


n 


n 


(40) 


Hence  the  conditions  for  the  existence  of  the  symmetrical  periodic  solutions 
in  question  can  be  satisfied  only  when  a  or  co  is  rational.  These  quanti- 
ties, given  in  (25),  are  power  series  in  e  and  they  depend  upon  M  and  /z0  and 
the  way  /*  is  generalized  when  the  transformation  JU  =  MO+^  is  made.  Since 
a  and  to  are  continuous  functions  of  /z,  MO,  and  e,  the  rationality  of  at 
least  one  of  them  at  a  time  can  be  assured.  It  should  be  noted  further  that 
<TO  and  co0  depend  upon  n,  ju0,  and  the  mode  of  generalization  of  n,  but  that 
they  are  independent  of  e . 

In  any  case  |X|  can  be  taken  so  small  that  the  series  will  converge,  but 
the  periodic  solution  does  not  belong  to  the  physical  problem  except  when 
X  =  ju  —  MO-  Suppose  the  series  diverge  for  this  value  of  X.  Theoretically 
the  values  of  the  coordinates  for  this  value  of  X  can  be  obtained  by  analytic 
continuation  with  respect  to  X  as  the  argument  from  the  periodic  solution 
which  exists  for  a  smaller  value  of  X.  There  is  an  exception  only  if  the 
function  has  a  natural  boundary,  or  if  X  =  ju  —  MO  is  a  singular  point. 


OSCILLATING    SATELLITES,    ELLIPTICAL   CASE.  233 

112.  The  Existence  of  Three-Dimensional  Symmetrical  Periodic 
Orbits.  Suppose  u-  is  the  rational  number  w  =  A'//i,  where  A  and  n  are 
relatively  prime  integers,  an<l  take  T  =  2me.  Suppose  aT  is  not  an  integral 
multiple  of  2r.  Then  A,^0  and  the  first  four  equations  of  (31)  can  be 
solved  for  a,,  .  .  .  ,  a,  uniquely  as  power  series  in  c,,  c,,  and  X,  vanishing 
identically  for  <•,  =  <•,  =  (),  by  property  [11].  The  a,  ,  .  .  .  ,  a4  are  even 
functions  of  c,  and  c,  taken  together,  by  property  [9].  When  these  results 
are  substituted  in  the  last  two  equations  of  (31),  they  become  power  series 
in  c,  ,  c,  ,  and  X.  These  series  are  of  odd  degree  in  c,  and  c,  taken  together, 
by  proj>erty  [10],  and  therefore  vanish  identically  for  c,  =  c,  =  0.  The 
substitution  of  the  values  for  a,  ,  .  .  .  ,  a4  does  not  change  the  linear  terms. 
for  the  first  four  equations  were  even  in  c,  and  c,  alone. 

Let  r,  be  eliminated  by  means  of  the  fifth  equation  of  (31).  Then 
r,  is  a  factor  of  the  result,  which  has  the  form 

0  =  cl[aMX+ax(^+  •  •  •  ].  (41) 

The  solution  c,  =  0  is  trivial  and  we  are  interested  only  in  those  obtained  by 
setting  the  other  factor  equal  to  zero.  The  second  factor  set  equal  to  zero 
is  satisfied  by  r,  =  X  =  0.  If  a,,  is  distinct  from  zero,  solutions  for  c,  as  power 
series  in  fractional  powers  of  X  certainly  exist.  If  a/l0  is  the  first  alo  which 
does  not  vanish,  then  the  solutions  are  expansible  as  power  series  in  X!//1. 
In  particular,  if  a*,  is  distinct  from  zero  the  solutions  are  expansible  as 
power  series  in  ±X*.  If  the  number  of  solutions  is  odd,  only  one  is  real;  and 
if  even,  only  two  are  real,  and  these  are  real  only  for  positive  or  negative 
values  of  X  according  as  a,,  and  a/|0  are  unlike  or  like  in  sign. 

It  is  clear  a  priori  that  the  number  of  solutions  will  be  even,  for  there 
is  nothing  of  a  dynamical  nature  by  which  to  distinguish  the  two  sides  of 
the  xi/-plane.  Consequently,  if  any  initial  projection  gives  rise  to  a  periodic 
orbit,  a  symmetrically  opposite  one  with  respect  to  the  ary-plane  will  also 
produce  a  periodic  orbit. 

It  remains  to  show  that  a,,  is  distinct  from  zero.  To  prove  this  the 
terms  of  the  second  degree  in  the  a,,  ct,  and  X  must  be  considered  (§108). 
It  follows  from  the  form  of  (29)  and  (30)  that  o^  depends  only  upon  the 
z-equation,  for  it  is  not  changed  by  the  substitution  of  the  solutions  of  the 
fir<t  four  equations  of  (31)  for  a,  ,  .  .  .  ,  a4  in  the  last  two.  Hence  a,, 
depends  only  upon  the  solution  of 

z"+[A+3Aecost+  •  •  -]z,=  [A'+3A'ecost+  •  •  -}zt,          (42) 
where 


It  follows  from  the  properties  of  wlt  wt,  u  and  the  general  theory  of  §§29 
and  30  that  the  coefficient  o^,  is  a  power  series  in  e.  For  e  =  0  it  was  given 
in  equation  (26),  of  Chapter  VI,  where  it  was  shown  to  be  (-l)'A<rA'.  This 
being  distinct  from  zero,  e  can  be  taken  so  small  that  the  scries  for  a,,  is 
distinct  from  zero. 


234  PERIODIC    ORBITS. 

Similarly,  o,,0  is  made  up  of  a  constant  part  distinct  from  zero  plus  a 
converging  series  in  e,  and  is  therefore  distinct  from  zero  for  e  sufficiently 
small.  Suppose  e  =  0  in  (41)  and  let  the  value  of  Cj  obtained  from  solving 
the  resulting  equation  be  c{0>;  then  let  c1  =  cf)(l+7).  It  is  easily  found 
from  (41)  that  dy/de  is  a  power  series  in  e  and  7  which  is  distinct  from 
zero  for  7  =  e  =  0  provided  X  has  such  a  value  that  cf'^0.  Hence  the 
solution  of  (41)  can  be  written  in  the  form 


(43) 

where  p(±\*)  is  a  power  series  in  X*  whose  coefficients  are  power  series  in  e. 
On  substituting  this  result  in  the  solutions  of  the  first  four  equations  of  (31) 
for  the  at  as  power  series  in  ct  and  X,  we  have  c^  ,  .  .  .  ,  a4  expressed  as  power 
series  in  X*.  But  since  ax  ,  .  .  .  ,  a4  contain  only  even  powers  of  Cj  ,  they  have 
X  instead  of  X*  as  a  factor  after  Cj  is  eliminated  by  (43).  The  expressions 
for  the  coordinates  become,  since  x  and  y  are  functions  of  c\  , 

x  =  \Pl(\*;t),  y  =  XP,(X»;0,  2  =  X*P,(X*;  0,  (44) 

where  P,  ,  P2,  and  P3  are  power  series  in  X*. 

Since  the  problem  is  dynamically  symmetrical  with  respect  to  the 
rc?/-plane,  a  solution  symmetrically  opposite  with  respect  to  the  xy-pl&ne 
exists  for  all  |X|  sufficiently  small.  Therefore  z  must  be  an  odd  series  in 
XJ,  and  x  and  y  even  series  in  X*.  Hence  equations  (44)  become 

*  =  XQ1(X;0,  y  =  XQ,(X;0,  s  =  X»Q,(X;  0,  (45) 

where  Q1(  Q2,  and  Q3  are  power  series  in  X. 

We  suppose  that  e>0  and  therefore  that  the  finite  bodies  are  at  their 
perifoci*  at  t=0.  The  infinitesimal  body  crosses  the  re-axis  perpendicularly 
at  t=0  and  at  t=T/2.  It  follows  from  the  symmetry  of  the  motion  that 
it  can  cross  the  re-axis  perpendicularly  only  at  the  end  and  middle  of  the 
true  period.  Therefore  if  n  and  N  are  relatively  prime,  T  =  2mr  =  2Nir/u 
is  the  true  period. 

The  two  cases  I,  n  even,  and  II,  n  odd,  merit  a  little  further  discussion. 
In  the  first  N  is  odd  because  n  and  N  are  relatively  prime;  in  the  second  it 
may  be  either  odd  or  even. 

113.  Case  I,  n  even,  N  odd.  —  The  infinitesimal  body  crosses  the  re-axis 
perpendicularly  at  t  =  0  and  also  at  t  =  T/2  =  mr  =  2n'ir,  where  n'  is  an 
integer.  Since  at  both  of  these  epochs  the  finite  bodies  are  at  their  peri- 
foci,  the  infinitesimal  body  crosses  the  re-axis  perpendicularly  only  when  the 
finite  bodies  are  at  their  perifoci.  It  follows  from  this  that  the  infini- 
tesimal body  crosses  the  re-axis  perpendicularly  at  the  same  point  but  in 
the  opposite  direction  with  respect  to  the  rn/-plane  at  t=0  and  t  =  T/2.  To 

*Perifoci  will  be  used  to  denote  the  points  at  which  the  finite  bodies  are  nearest  each  other,  and 
apofoci  those  at  which  they  are  moat  remote  from  each  other.  These  points  correspond  to  perihelia 
and  aphelia  in  planetary  motion. 


O8CILLATINC!    SATKI  I  I  I  l>.   KI.I.IIMICAL   CA  235 

prove  it  Mippo.M-  tin-  two  points  were  different.  Then  we  should  li:i\c  four 
solutions  corresponding  to  ±X*  at  each  of  the  points,  and  it  is  known  that 
there  are  but  two.  The  values  of  z'(0)  and  z\T/2)  are  opposite  in  sign 
because  othriwi-r  the  period  of  the  motion  would  be  T/2. 

It  can  now  be  shown  that  in  this  case  the  orbits  obtained  by  taking 
the  two  signs  before  X*  are  geometrically  the  same  one.  Consider  the 
orbit  defined  by  the  positive  sign  before  X*.  At  the  time  t  =  T/2  the 
infinitesimal  body  crosses  the  x-axis  perpendicularly  at  the  point  at  which 
it  crossed  at  t  =  Q,  but  in  the  opposite  z-direction.  We  may  consider  T/2 
as  an  origin  of  time  for  defining  orbits  which  cross  the  x-axis  perpendicu- 
larly. It  has  been  shown  that  there  is  but  one  with  the  given  z-direction 
of  motion,  and  at  t=  T/2  +  T/2  =  T  the  infinitesimal  body  will  again  cross  the 
x-axis  perpendicularly  with  the  opposite  z-direction,  viz.,  with  that  which 
it  had  at  J  =  0.  Since  this  orbit  was  unique,  it  follows  that  the  two  orbits 
which  correspond  to  the  double  sign  before  X1  are  geometrically  the  same, 
but  that  in  one  the  infinitesimal  body  is  half  a  period  ahead  of  its  position 
in  the  other  one.  That  is,  changing  the  sign  of  A'  in  the  solution  is 
equivalent  to  adding  T/2  to  t. 

When  the  solutions  are  actually  constructed  and  are  reduced  to  the 
trigonometric  form,  they  involve  sines  and  cosines  of  (ju  +  k)t,  where,; 
and  A;  are  integers.  Since  x,  y',  and  z'  are  even  functions  of  t,  they  will 
involve  only  cosines,  and  since  x',  y,  and  z  are  odd  functions  of  t,  they 
will  involve  only  sines.  Since  x  and  y  are  even  series  in  X',  it  follows  from 
the  foregoing  properties  that  in  them 


These  identities  are  satisfied  if,  and  only  if, 

cosO'w+A;)  T/2  =  cos(ju+k)2irnf=  1     (nf  an  integer); 

therefore 

N 
(j<ji+k)ri=j  y  +  kn'  =  p    (p  an  integer). 

Since  N  is  odd  it  follows  from  this  relation  that  j  is  necessarily  even. 
Similarly,  in  the  case  of  the  series  for  z  the  relation 

siu[(ju+k)t]  =  -  sin[(ju+k)(t+T/2)] 
must  lie  fulfilled.     It  follows  from  this  identity  in  t  that 

(ju+k)n'  =  j  ^  +  kn'  =  2-£±±    (p  an  integer). 

Tliis  relation  can  be  satisfied  only  if  j  is  an  odd  integer. 

There  are  solutions  in  this  case  which  have  not  been  obtained  by  the 
analysis  as  given.  The  orbits  which  have  been  discussed  intersect  the  x-axis 
perpendicularly  when  the  finite  bodies  are  at  their  perifoci,  and  obliquely. 
if  at  all.  when  they  are  at  their  apofoci.  Similarly,  supposing  t  =  0  when  the 


236  PERIODIC    ORBITS. 

finite  bodies  are  at  their  apofoci,  it  can  be  proved  that  there  are  orbits 
with  the  same  period  in  which  the  infinitesimal  body  crosses  the  re-axis  per- 
pendicularly when  the  finite  bodies  are  at  their  apofoci,  and  obliquely,  if 
at  all,  when  they  are  at  their  perifoci. 

1 14.  Case  II,  n  odd. — In  the  present  case  the  infinitesimal  body  crosses 
the  z-axis  perpendicularly  when  the  finite  bodies  are  at  their  perifoci  and 
also  when  they  are  at  their  apofoci,  because  T/2  is  an  odd  multiple  of  TT. 
If  N  is  even,  the  solution  for  +  X*  is  geometrically  distinct  from  that 
for  —  X*.  To  prove  this,  we  note  that  |X|  can  be  taken  so  small  that  the 
sign  of  the  series  for  z'  is  determined  by  its  first  term.  If  e  is  small,  z'  has 
its  sign  determined  by  the  constant  parts  of  wt  and  w2 .  Now  the  first 
parts  of  these  first  terms,  viz.,  e"^1'  and  e~uv^',  have  the  period  2ir/u. 
Since  T=  2Nir/u,  and  N  is  an  even  integer,  it  follows  that  the  sign  of  z'  is 
the  same  at  2  =  0  and  t=T/2.  Now  suppose  X  to  increase  to  the  value 
belonging  to  the  physical  problem.  If  the  sign  of  z'  (T/2)  changes  it  must 
do  so  by  passing  through  zero.  But  in  this  case,  since  z(T/2)  is  also  zero, 
z  would  be  identically  zero.  Therefore  z'  (0)  would  also  be  zero  and  would 
change  sign  for  the  same  value  of  X,  and  z'  would  still  have  the  same  sign 
at  t  =  0  and  t  =  T/2.  Consequently,  the  z-component  of  velocity  at  i  =  T/2 
is  not  equal  to  the  negative  of  that  at  t  =  0.  Hence,  when  n  is  odd  and  N 
is  even,  the  orbits  for  +XJ  and  —  X*  are  geometrically  distinct,  because  a 
single  orbit  can  cross  the  z-axis  perpendicularly  but  twice. 

If  n  and  N  are  both  odd,  the  infinitesimal  body  crosses  the  z-axis  per- 
pendicularly, as  before,  both  when  the  finite  bodies  are  at  their  perifoci 
and  also  when  they  are  at  their  apofoci.  But  though  in  this  case  the  sign 
of  z'  at  t=Q  is  opposite  to  that  at  t  =  T/2,  the  orbits  for  +X}  and  -X*  are  dis- 
tinct; for  otherwise  the  motion  of  the  infinitesimal  body  would  be  precisely 
the  same  while  the  finite  bodies  were  moving  from  perifoci  to  apofoci  as 
while  they  were  moving  from  apofoci  to  perifoci.  This  is  impossible 
because  t  enters  the  differential  equations  differently  in  the  two  cases. 

If  we  set  up  the  problem  taking  t  —  0  when  the  finite  masses  are  at  their 
apofoci,  we  shall  find  similarly  two  solutions;  but  they  will  be  identical 
with  these,  for  in  these  the  infinitesimal  body  crosses  the  z-axis  perpen- 
dicularly when  the  finite  bodies  are  at  their  apofoci. 

The  numbers  n  and  N  have  so  far  been  taken  relatively  prime,  and 
T=2nir.  We  shall  now  inquire  whether  there  are  other  solutions  with  the 
period  T'  =  nT,  where  K  is  an  integer.  The  determinant  Aj,  vanishes  for  this 
value  of  t.  Solutions  having  this  period  certainly  exist,  for  they  include  as 
special  cases  those  with  the  period  T.  Proceeding  as  in  finding  equation 
(41),  the  corresponding  steps  are  taken,  and  it  is  found  that  in  this  case  a01 
and  a20  differ  from  their  former  values  only  by  multiples  of  2ir.  Therefore  the 
number  of  solutions  with  the  period  T'  is  the  same  as  with  the  period  T.  Hence 
there  are  no  new  solutions  whose  periods  are  multiples  of  those  considered. 


»>i  -ILL.  MINI;  >Mi:i,i.m>.  Ki.i.ii'neAL  i:.\  237 


115.  Convergence.  The  existenceof  the  >\  mmetrical  three  diniriisional 
periodic  orbits  has  been  proved  except  for  a  discussion  of  the  convergence 
of  the  series  which  have  been  employed,  a  matter  which  must  now  he  taken 
up.  The  nature  of  the  difficulty  will  first  be  pointed  out.  Equations  (13) 
were  integrated  a>  POWIT  -erics  in  the*/,,  c,,  and  X.  It  was  shown  in  §§14-16 
that  for  any  prea-Mum-d  T  the  moduli  of  these  parameters  can  tie  taken  so 
small  that  the  series  converge  for  all  0^<^  T.  The  limits  on  the  moduli  of 
the  a,,  ct,  and  X  are  functions  of  /*,  M3,  and  e.  But  T  can  not  be  taken 
arbitrarily  in  advance,  for  it  is  a  discontinuous  function  of  u,  which  is  in 
turn  a  function  of  n,  MO,  and  c.  It  is  not  evident  a  priori  that  values  of 
M,  MO,  and  e,  satisfying  the  relations  M  —  MO  =  X,  W=/(M,  MO,  e),  exist  such  that 
all  the  series  which  are  employed  are  convergent. 

The  final  parameters  of  the  solutions  are  M,  M»>  e,  and  the  mode  of 
generalizing  M  is  arbitrary.  Suppose  the  ratio  of  the  finite  masses  is  given, 
that,  is,  that  M  is  a  fixed  number.  Suppose  also  that  the  mode  of  general- 
izing M  has  been  determined.  It  will  be  shown  that  values  of  MO  and  e  exist 
such  that  the  series  all  converge  for  X  =  M—  MO- 

In  equations  (25)  it  was  shown  that 


where  \/A  and  the  w,  are  functions  of  M».  Moreover,  a  detailed  examina- 
tion of  the  functional  relation  shows  they  are  continuous  functions  of  MO- 
Suppose  for  any  MO  such  that  |M—  Mo|^«i>0  the  series  for  w  converges  for 
M  ^  »?i  ,  where  »?,  depends  on  «,  .  It  is  easy  to  show  from  the  nature  of  the 
dependence  of  the  series  for  w  upon  M  and  MO  that  n,  >0.  Take  any  particu- 
lar MB"  such  that  |M  —  Mi"l^«i  and  suppose  that,  while  e  runs  over  the  range 
0  to  T;,.  the  value  of  w  (MO,  e)  runs  over  the  range  «  (MO",  0)  =  VA  to  «  (MO",  fi)- 
For  brevity,  let  w<0)  and  w(U  represent  the  smallest  and  largest  values  of  w. 
An  examination  shows  that  «,,  u>t,  .  .  .  are  not  all  identically  zero,  and  it 
follows  from  this  that  |  co(l)  -  w<0)  |  >  0.  Let  MO"  take  all  real  values  such 
that  |M  —  MO"!^*!  and  find  the  corresponding  values  of  w<0)  and  w'".  Let 
u™  be  the  greatest  «(0),  and  wj"  be  the  least  w(l>.  The  value  of  «,  can  be 
taken  so  small  that  w^-w^X),  and  hence  from  the  continuity  of  w  as  a 
function  of  MO  ''  follows  that  u  takes  all  values  satisfying  the  inequalities 
«"'  ^w  ^w,(l>.  Take  any  rational  value  of  w  depending  on  M,  Mo  ,  and  e  which 
satisfies  these  inequalities,  say 

«o=^-°'  (46) 

n« 

where  Nt  and  n,  are  relatively  prime  integers.  Then  determine  T=  Tt  by 
the  equation 

(47) 


238  PERIODIC    OKBITS. 

Now  consider  the  series  (31),  in  which  we  put  T=  T0.  Since  they  vanish 
identically  for  a,  =  c,  =  0,  the  discussion  in  §§14-16  shows  that,  for  any  values 
of  X  and  e  for  which  the  differential  equations  are  regular,  rt>  0  and  p4>0 
can  be  so  determined  that  the  series  converge  for  all  0  <:  T  ^  T0  provided 
|a4|<r,,  |C(|<P(.  The  rt  and  pt  are  functions  of  MO>  \  and  e.  We  may 
eliminate  X  by  the  relation  ^  =  /z0+X,  and  we  shall  take  |/u—  MO  I  =ei  and 
0  ^  e  i=  r]1  ,  where  et  and  ^  are  both  distinct  from  zero  and  have  such  values 
that  the  differential  equations  are  regular  for  M—  Mol  =ei  >  e  <^i  •  Let  r™  and  p,<0) 
be  the  least  values  of  rt  arid  p,  as  MO  and  e  take  all  values  satisfying  the  ine- 
qualities |M  —  Mol  =£i>  0  <S  e  =  *7i-  It  has  been  shown  that  when  the  solutions 
of  (31)  exist  they  have  the  form 

a,  =  Xp,(Xi,  e)  Cj=  -Cg  =  X}p5(X},  e),  (48) 

where  the  p<  are  power  series  in  X!  and  e.  These  series  will  converge  and 
give  |a,|<r(<0),  |c,|<pf  provided  0<|X|^e,^eM  for  all  e<r?,.  That  is, 
since  the  series  vanish  identically  for  X  =  0,  the  limits  on  the  a{  and  c,  can  be 
controlled  by  X  alone.  Choose  any  X  satisfying  the  inequality  |X|^e2  and 
determine  MO  by  the  relation  M~MO  =  ^-  It  was  shown  above  that  for  this 
Mo  there  exists  an  e  <  ^  such  that  co  (MO  ,  e)  —  w0  =  N0/na  .  Hence  for  this  MO 
and  e  all  the  series  employed  are  convergent;  that  is,  the  existence  of  certain 
solutions  of  the  type  in  question  is  proved. 

The  question  might  be  asked  whether  the  solutions  exist  if  both  M  and  e 
are  given  in  advance.  It  is  not  easy  to  make  the  answer  in  general,  but 
it  is  clear  that  the  mode  of  generalization  of  M  into  M  and  MO  +  X  opens  a 
wide  range  of  possibilities.  This  is  so  unless,  indeed,  the  realm  of  validity 
of  the  results  is  independent  of  this  process.  Suppose  M,  c,  and  the  initial 
conditions  are  given  such  that  the  motion  is  periodic.  The  coordinates 
may  be  represented  by 


Consider  the  expansions  of  the  functions  Ft  as  power  series  in  X,  where 
M  =  MO+^  m  at  least  part  of  the  places  in  which  M  occurs.  It  is  clear  that  the 
realm  of  convergence  of  the  series  depends  upon  the  manner  of  this  trans- 
formation. For  example,  if  F1  =  Ft  =  F3—  sinZ/(l+  X)(2+/t),  and  if  we  put 
l+M=l+Mo+^j  2+M  =  2+M,  then  the  functions  are  expansible  as  power 
series  in  X  which  converge  if  |X|  <!+/*„,  where  X  and  MO  are  subject  to  the 
condition  MO+^  =  M-  But  if  we  put  I+M=I+M>  2+M  =  2+M0+X,  then 
the  series  in  X  are  convergent  if  |X|  <2+M0  where  X  and  MO  are  subject  to  the 
condition  MO  +  ^=M-  For  example,  if  M  =  l/3  in  the  first  case  the  series 
converge  if  |X|<2/3,  and  in  the  second  case  if  |MJ<7/6.  Now  it  is  clear 
from  the  variety  of  ways  in  which  X  can  be  introduced  that  convergence  of 
the  series  can  be  secured  in  many,  if  not  all,  cases  when  M  and  e  are  given 
in  advance. 


OBCII.I.UIV,    -\  I  1  .1.1.111.-.    Ll.l.  11MICAL    CA  239 

1  16.  The  Existence  of  Two-Dimensional  Symmetrical  Periodic  Orbits. 
The  last  two  equations  of  (31)  are  satisfied  identically  by  c,  =  c,  =  0,  in  which 
case  the  orbits  become  plane  curves.     We  have  to  consider  the  solution  of 
the  first  four  equations  for  a,  ,  .  .  .  ,  a,  in  terms  of  X. 

The  determinant  A,  equation  (33),  now  becomes  simply  A  =  A,.  It 
follows  from  i.'M)  that  the  condition  Ai=0  can  be  satisfied  only  by  <r=N/n, 
\\  here  A"  and  n  are  integers  which  we  shall  suppose  are  relatively  prime. 

\\  V  shall  solve  the  first  three  equations  of  (31)  for  a,,  a,,  and  a,  as 
power  series  in  a,  and  X.  This  solution  exists  and  is  unique  provided  the 
determinant  of  the  coefficients  of  the  linear  terms  in  a,,  a,,  and  a4  is 
distinct  from  zero.  It  follows  from  (34)  that  this  determinant  is 


which  is  distinct  from  zero  unless  As  is  zero. 

If  A,  were  zero  we  should  use  the  first,  second,  and  fourth  equations  of 
(31).  The  determinant  of  the  coefficients  of  the  linear  terms  of  a,,  at, 
and  a,  in  these  equations  is  found  from  (34)  to  be 


which  is  distinct  from  zero  unless  vt(T/2)  is  zero.  But  At  and  v}(T/2)  can 
not  both  vanish,  for  then  Z),  of  (37)  would  be  zero,  whereas  it  is  distinct 
from  zero.  Therefore  a,,  a,,  and  a4  can  always  be  eliminated  by  means 
of  three  of  equations  (31),  leaving  a  single  equation  of  the  form 

0  =  a,[/3MX+Aoa1+/311a1X+/310a1*+      •  •]•  (49) 

This  equation  carries  a,  as  a  factor  because  equations  (31)  are  identically 
satisfied  by  a,=  •  •  •  =a4  =  0. 

Now  consider  equation  (49).  The  trivial  solution  a,  =  0  will  be  rejected, 
and  we  shall  attempt  to  solve  for  o,  as  power  series  in  X'",  where  j  is  an 
integer.  If  001  is  distinct  from  zero  such  a  solution  certainly  exists,  and  j 
is  determined  by  the  first  &„  which  is  distinct  from  zero.  The  coefficient  ft, 
is  a  power  series  in  e  whose  term  independent  of  e  was  found  in  Chapter  VI, 
equation  (26),  to  be  distinct  from  zero.  Consequently  for  e  sufficiently 
small  /3OT  is  distinct  from  zero,  and  the  solutions  exist. 

Now  consider  0IO  .  It  was  shown  in  Chapter  VI,  §  100,  that  the  part 
of  this  coefficient  independent  of  e  is  zero.  It  will  now  be  shown  that  it  is 
identically  zero.  The  solutions  of  the  first  three  equations  of  (31)  for 
a,,  a,,  and  a,  contain  no  terms  of  the  first  degree  in  a,  alone,  for,  if  wo 
eliminate  a,  and  a,  by  means  of  the  first  two  equations,  we  have 

At   ep^  -  e~P*   a4  =  0+  terms  of  second  and  higher  degrees.         (50) 


It  follows  from  property  [3]  of  §  108  that  the  terms  in  the  solutions  of  the  second 
degree  in  the  a,  do  not  contain  t  as  a  factor.  Hence,  the  only  terms  of  the 
second  degree  in  the  o<  which  are  not  periodic  with  the  period  T  are  those 


240  PERIODIC   ORBITS. 

which  contain  al  or  a\  as  a  factor.  Hence  the  solution  (50)  for  a4  has  a 
term  of  at  least  the  third  degree  in  o,  as  the  term  of  lowest  degree  in  at 
alone.  Consequently,  when  the  first  three  equations  of  (31)  are  solved  for 
a2,  a3,  and  a^  and  substituted  in  the  last  one,  the  result  contains  no  term 
of  the  second  degree  in  a,  .  That  is,  |8IO  is  identically  zero. 

It  is  not  necessary  to  consider  ft,  if  /?,„  is  distinct  from  zero.  In  Chapter 
VI,  §100,  it  was  shown  that  the  part  of  &20  which  is  independent  of  e  is 
distinct  from  zero.  It  is  also  distinct  from  zero  for  e  sufficiently  small. 
Therefore  the  solutions  arc  expansible  as  power  series  in  ±X*  of  the  form 


One  value  of  the  double  sign  belongs  to  the  orbit  when  the  infinitesimal 
body  crosses  the  axis  in  one  direction,  and  the  other  when  it  crosses  it  in 
the  other  direction. 

There  are  two  cases  to  be  considered,  according  as  n  is  even  or  odd  in 
the  expression  a  =  N/n. 

117.  Case  I,  n  even.  —  The  infinitesimal  body  crosses  the  .r-axis  perpen- 
dicularly at  t  =  Q  and  at  t=T/2  =  mr  =  2nfTr,  where  n'  is  an  integer.  Since 
2n'ir  is  an  integral  multiple  of  the  period  of  revolution  of  the  finite  bodies, 
and  since  the  infinitesimal  body  crosses  the  z-axis  perpendicularly  only  at 
the  beginning  and  middle  of  the  period,  it  follows  that  it  crosses  the  z-axis 
perpendicularly  only  when  the  finite  bodies  are  at  their  perifoci.  It  follows, 
as  in  the  case  of  the  three  dimensional  orbits,  that  the  orbit  belonging  to  —  X* 
is  not  geometrically  distinct  from  that  belonging  to  +X};  in  particular,  that 
one  orbit  can  be  obtained  from  the  other  by  increasing  t  by  T/2. 

Consider  the  terms  which  are  even  in  X1.  They  are  not  altered  by 
changing  the  sign  of  X5.  Consequently  in  these  terms 


sin 
cos 


from  which  it  follows  that  j  is  an  even  integer.     In  the  terms  of  odd  degree 
in  X!  we  have 


sin 
cos 


from  which  it  follows  that  j  is  an  odd  integer. 

If  we  should  set  up  the  problem  starting  the  infinitesimal  body  per- 
pendicularly to  the  x-axis  when  the  finite  bodies  are  at  their  apofoci,  we 
should  find  similarly  two  geometrically  identical  orbits  in  which  the  infini- 
tesimal body  crosses  the  z-axis  obliquely  when  the  finite  bodies  are  at  their 
perifoci.  That  is,  when  n  is  even  there  are  two  classes  of  geometrically 
distinct  orbits  of  given  period  which  intersect  the  :r-axis  perpendicularly;  in 
one,  the  periodic  orbits  intersect  it  thus  only  when  the  finite  bodies  are  at 
their  perifoci,  and  in  the  other  only  when  they  are  at  their  apofoci. 


"-'  II,I,ATIN<.    ^\l  I.1.I.I1K>.    KI.UPTICAL   CASE.  241 

1  18.  Case  II,  n  odd.  If  in  the  case  where  n  is  odd  the  infinitesimal  body 
crosses  the  .r-axis  perpendieuhirly  at  i  =  ()  and  the  finite  bodies  arc  at  (heir 
perifoci,  then  it  also  crosses  the  .r-axis  perpendicularly  at  i^T/'l  \\hen  the 
finite  bodies  arc  at  their  apofoci.  If  N  is  even,  the  infinitesimal  body 
ero»cs  the  ./--axis  in  the  same  direction  at  t  =  ()  and  J  =  7'/2.  Hence  the 
orbits  for  +  X*  and  —  X!  are  in  this  case  geometrically  distinct. 

If  N  is  odd,  the  orbits  for  +X'  and  —X*  are  also  distinct,  because  other- 
ui.-ethe  mot  ion  of  the  infinitesimal  body  would  be  the  -a  me  while  the  finite 
bodies  are  moving  from  perifoci  to  apofoci  as  while  they  are  moving  from 
apofoci  to  perifoci.  This  is  impossible  because  /  enters  the  differential 
equations  differently  in  the  two  cases. 

Similarly,  starting  when  the  finite  bodies  are  at  their  apofoci,  two  geo- 
metrically distinct  orbits  arc  obtained,  in  both  of  which  the  infinitesimal 
body  crosses  the  .r-axis  perpendicularly  when  the  finite  bodies  are  at  their 
apofoci.  and  also  \\hen  they  are  at  their  perifoci.  These  orbits,  therefore, 
are  identical  with  those  obtained  starting  when  t  he  finite  bodies  were  at  their 
perifoci. 

CONSTRUCTION  OF  THREE-DIMENSIONAL  PERIODIC  SOLUTIONS. 

119.  Defining  Properties  of  the  Solutions.  —  It  has  been  shown  that 
the  periodic  solutions  have  the  form 

x=  2  xtl\',          y=  2  y,,\',          z=  2  z^-.X"2*",  (51) 

/-i  /-i  i-\ 

whore  the  x,,  yn  and  zt  are  all  periodic  with  the  period  2T/w.  It  has  been 
shown  that  symmetrical  orbits  exist,  two  for  each  value  of  X,  and  their 
coefficients  are  uniquely  determined  by  the  periodicity  conditions  and 
2(0)  =0.  It  follows  from  this  last  relation  and  from  the  fact  that  the  series 
for  x,  y,  and  z  converge  for  all  |  X  |  sufficiently  small,  that  each  z,(0)  sepa- 
rately is  zero. 

120.  Coefficient  of  X'.  —  It  follows  from  (13)  that  this  term  is  defined  by 

,  =  (>.  (52) 


This  equation  is  of  the  type  of  that  treated  in  §53,  and  its  general  solution 
is  of  the  form 

z,  =  c|ue"^'wi(«;  0  +cjl)e-"^'  w,(e;  t),  (53) 

where 


while  wt  differs  from  wl  only  in  the  sign  of  N/^M,  and  where  each  ir{"  is 
separately  periodic  with  the  period  2r.  Since  (53)  is  unchanged  by  changing 
the  signs  of  both  t  and  v^T,  it  follows  that  in  the  expressions  for  u>,  and  wt 


242  PERIODIC    ORBITS. 

the  coefficients  of  the  cosine  terms  are  real,  and  those  of  the  sine  terms  are 
purely  imaginary.  Since  w^e;  0)  =  w2(e;  0)  =  1  for  all  values  of  e,  it  follows 
from  the  condition  z,(0)  =  0  that  c[l)+c?  =  Q. 

It  will  now  be  shown  that  w  is  a  series  in  even  powers  of  e.  The  coeffi- 
cient of  Zj  in  (52)  is  derived  from  the  expansion  of  r^3  and  r^3.  Now  the 
expressions  for  r,  and  r2  are  unchanged  if  in  them  e  is  replaced  by  —  e  and 
t  by  t+w.  Consequently,  if  efaMvrriiw1(e',  t)  is  a  solution  of  (52),  then  also 
ew(~e)v/^T('+T)t01(  —  e;  t+w)  is  a  solution.  Since  any  solution  can  be  expressed 
linearly  in  terms  of  the  two  solutions  in  (53),  and  since  the  solutions  now 
under  consideration  hold  for  all  e  sufficiently  small,  and,  for  e  =  0,  differ  only 
by  the  factor  e?'^^1**,  it  follows  that  for  any  e  they  differ  only  by  a  constant 
factor.  Therefore 


*Wi  (e;  t)  -  Ce^^^w^-e;  t+ir)  =  0. 

This  relation  is  satisfied  identically  in  t  and  e.     Since  wl  is  periodic  with 
the  period  2^,  it  follows  also  that 


Hence  we  have  two  homogeneous  linear  equations  in  e^^^'w^e;  t)  and 
Crew(~"v^T('+'r)w;1(—  e;  t+ir),  and  as  these  quantities  are  not  identically  zero, 
the  determinant 


must  vanish.     Since  by  definition  o>  must  reduce  to  VA  for  e  =  0,  we  have 
w(  —  e)=u(e);  that  is,  w  is  a  function  of  e2. 

Upon  carrying  out  the  computation  by  the  method  of  §53,  we  find 


(54) 


_  9A(1-3A-2A')        2,_ 
8(1-A)(1-4A)C 

It  will  be  noticed  that  the  coefficients  of  cosjt  and  sinjt  have  ej  as  a  factor 
so  far  as  they  are  written.  This  is  a  general  property  of  the  differential 
equations,  and  an  examination  of  the  process  of  integration,  as  explained 
in  §53,  shows  it  is  also  a  general  property  of  the  solutions. 

If  we  had  solved  the  differential  equations  for  x^  and  yl ,  we  should  have 
found  that  these  quantities  are  identically  zero  because  of  the  periodicity 
conditions  to  which  the  solutions  are  subject. 


OSCILLATING    SATELLITES,    ELLIPTICAL   CASE.  243 

121.  Coefficients  of  ,\.     It  follows  from  (13)  and  (14)  that  these  term- 
are  defined  bv 


(66) 


,-    [l-,4-34ecos(+ 
where 


Jzj, 


the  signs  in  B  l)eing  the  first,  second,  or  third  pair  according  as  the  point 
(a),  (6),  or  (c)  is  in  question.     By  means  of  (53)  and  (54),  we  find 


\ 
A,- 


The  character  of  the  solutions  of  equations  of  the  type  to  which  (55) 
belong  was  discussed  in  §29.  It  was  there  shown  that  they  consist  of  the 
complementary  function  plus  terms  of  the  same  character  as  Xt  and  F,. 
It  follows  from  the  hypothesis  that  the  imaginary  characteristic  exponent 
irV  —  1,  arising  in  the  solution  of  (55),  and  wV—  I  are  incommensurable, 
that  the  constants  of  integration  must  all  be  put  equal  to  zero.  The  par- 
ticular integrals  can  be  found  most  conveniently  by  assuming  their  form 
with  undetermined  coefficients,  and  then  defining  them  by  the  conditions 
that  the  equations  shall  be  identically  satisfied. 


244  PERIODIC   ORBITS. 

We  shall  need  the  following  properties  of  the  solutions  of  equations 
(55).  They  are  homogeneous  of  the  second  degree  in  Ci1)ewv/^7'  and 
ca)e-Wv^T(  The  terms  in  the  solutions  multiplied  by  (c2")2  differ  from  those 
multiplied  by  (c"')2  only  in  the  sign  of  V—  i,  because  this  is  a  property  of  the 
right  members  of  the  differential  equations.  If  throughout  equations  (55) 
we  change  y2  into  -yz,  V^i  into  -v^T,  and  t  into  -t,  the  equations  are 
unchanged.  Therefore,  in  the  expression  for  z2  the  coefficients  of  the  cosine 
terms  are  real  and  the  coefficients  of  the  sine  terms  are  purely  imaginary. 
The  opposite  is  true  for  ?/2  .  The  terms  in  x2  having  c"'^"  as  a  factor  involve 
only  cosines  and  are  independent  of  the  exponentials  e"^~11  and  e~a^^1,  while 
those  in  7/2  having  c"X"  as  a  factor  involve  only  sines  and  are  also  indepen- 
dent of  the  exponentials  e<"v/=T(  and  e~us/^T(.  It  follows  from  these  properties 
and  c[u  =  -  c2"  ,  that  a£(0)  =  &(0)  =  0. 

Certain  divisors  are  introduced  in  the  integration  of  (55).  When  the 
right  members  of  these  equations  are  omitted,  the  solutions  are  of  the  form 
(24)  .  On  using  the  method  of  the  variation  of  parameters,  we  find 

fl|   —         -r  (i=l>    •    •    •    >  *), 

where  Ft(t)  has  the  form  of  Xt  and  F2,  and  where  A  is  the  determinant 
of  the  fundamental  set  of  solutions  (24).  It  follows  from  the  principles 
of  §18  applied  to  this  case  that  A  is  constant.  The  expressions  Ft(t) 
contain  terms  of  the  types  given  in  (56).  Consequently  the  at  contain 
terms  of  the  types 


Aa,=  (Oz  je^^'lA,  cosjt+  ^f=\B1  s\njt~]dt 


or,  performing  the  indicated  integrations, 


@^e-2<ov^'  [  -2co  v^Tcos^+j  smjt] 

u  smjt]  + 


Therefore,  the  divisor  f  —  4w2  appears  in  terms  involving  t  in  the  form 
g-zw^ncosy^  an(j  ^e  (Jivisorj  in  those  involving  t  in  sinjt. 

It  was  seen  that  in  the  expression  for  z,  the  coefficients  of  cosjt  and 
sin  jt  carry  e1  as  a  factor.  Consequently  it  is  true  also  for  z2  ,  and  therefore 
for  the  A,,  B},  and  C,. 


OSCILLATING    SATELLITES,  ELLIPTICAL   CASE.  245 

122.  Coefficient  of  X1*.— It  is  found  from  (13),  (14),  and  (51)  that 


Z3 


+3Aecost+ 


(57) 


The  coefficients  of  *,,  xtzl  ,  and  z}  are  series  involving  only  cosines,  and 
the  coefficient  of  ytzt  is  a  series  involving  only  sines. 

Now  consider  the  solutions  of  (57).    The  general  solution  of  the  left 
member  set  equal  to  zero  is 


where  C,  and  C,  are  the  constants  of  integration.  By  the  method  of  the 
variation  of  parameters,  the  conditions  on  C,  and  C,  that  (57)  shall  be 
satisfied  when  its  right  member  is  included  are 


Upon  solving  (58)  for  C{  and  Cj,  we  get 

.e"^^1',  (59) 


where 


—  w  V  —  iwt+w't 


It  follows  from  the  results  of  §  18  that  in  this  case  A  is  constant,  and  since 
wt  and  wt  are  power  series  in  e  the  determinant  A  is  also  a  power  series  in  e. 

In  order  that  the  solutions  of  (57)  shall  be  periodic  with  the  period  T 
it  is  necessary  and  sufficient  that  the  right  members  of  (59)  contain  no 
constant  terms.  We  must  therefore  pick  out  the  terms  in  -wtZt  and 
WiZ,  which  are  constants  multiplied  by  erv~lt  and  e'"*^'  respectively,  and 
set  their  coefficients  equal  to  zero. 

Consider  the  first  term  of  Z,  .  Upon  substituting  the  value  of  z,  from 
(53),  we  see  that  the  constant  part  of  the  coefficient  of  e"^1'  coming  from 
-«;,£,  is  the  constant  part  of  the  product 

(60) 


246  PERIODIC    ORBITS. 

The  corresponding  part  of  the  coefficient  of  e~uv~'  coming  from  -\-wtZt  is 
the  constant  part  of 


-L- 


(61) 


It  follows  from  the  properties  of  w1  and  w>2  that  their  product  involves  only 
cosines  and  that  the  coefficients  of  their  product  are  all  real.  Hence  the 
constant  parts  of  (60)  and  (61)  are  power  series  in  e  which,  aside  from  the 
coefficients  c[l)  and  c"',  differ  only  in  sign,  and  the  parts  independent  of  e 
are  respectively 


Cl 


J_\ 
if1/ 


Consider  now  the  terms  coming  from  that  part  of  Z3  which  contains 
as  a  factor.     The  constant  parts  of  the  coefficients  of  eWv/=Tl  and  e~wV=lt 
in  the  products 


and 

+wlxiz1[3B+12Becost  •  •  •} 

respectively  are  required.     It  follows  from  the  properties  of  wl  and  u>2  that 

wl[3B+12Beco8t+  •  •  •] 
and 

w2[3B+12Becost+  •  •  •} 

differ  only  in  the  sign  of  V^-T,  which  is  a  factor  of  all  the  sine  terms, 
while  the  coefficients  of  the  cosine  terms  are  all  real.  These  products 
do  not  involve  the  exponentials  e"v/=T'  and  e~wv^T(.  In  the  product  z2  zl  the 
coefficients  of  euv^^1  and  e~uv=lt  are  multiplied  by  (0*4"  and  c"}  (c"1)2  respec- 
tively. It  follows  from  the  properties  of  x2  and  zr  that  in  this  product 
the  coefficients  of  (c^Yc^e"^^'  and  c{u  (^0)te-"^=T'  differ  only  in  the  sign  of 
V—  1,  which  is  a  factor  of  all  the  sine  terms,  while  the  coefficients  of  the 
cosine  terms  are  all  real. 

The  typical  terms  of  the  products  —  ivt[3B+  12Be  cost  •  •  •]  and  that 
part  of  the  product  x22,  which  contain  ewv/~"  as  a  factor  are  respectively 


The  corresponding  terms  from  +u>,  [3B  +  12  Be  cost  -\-  •  •  •}  and  the  part 
of  x^Zi  containing  g-^'^7'  as  a  factor  are  respectively 


OSCILLATING    SATELLITES,    ELLIPTICAL   CASE.  247 

The  constant  parts  of  tin1  products  of  these  terms  are  respectively 


Since  these  properties  hold  each  term  individually,  it  follows  that  the  con- 
stant parts  of  the  coefficients  of  euy'-~il  and  e~vt>=::Tlin  —  wtxtzt[3B+  •  •  -]and 

are  res  Actively  of  the  form 

+  •  •  •  ],  +cjB(Oi[<VK?,H-  •  •  •  J.      (62) 

The  corrosjK>nding  discussion  shows  that  the  same  result  is  true  for  the 
third  and  fourth  terms  of  Zt.  Therefore,  in  order  that  the  solutions  of 
(57)  shall  be  periodic,  we  must  impose  the  conditions 


0  =  A'c["  +L(c;i))'ci1),  0  -  Kc?>  +Lc;u(O',  (63) 

where  A'  and  L  are  constants  and  power  series  in  e.  It  follows  from  the 
corresponding  work  in  Chapter  VI  that  both  K  and  L  have  terms  independent 
of  e  which  are  distinct  from  zero.  The  solutions  of  (63)  are 

c<u=c«u  =  0)  K+Ld?c?  =  0.  (64) 

The  former  leads  to  the  trivial  solution  x=y=z=0  and  need  not  be  con- 
sidered. Since  cj"=  —  c("  the  latter  gives 

•  •  •],  (65) 

and  c["  and  cj"  are  determined  except  as  to  sign.  The  orbits  which  correspond 
to  the  two  values  of  c|"  are  geometrically  identical  or  distinct  according  as 
T  is  an  odd  or  even  multiple  of  2r  (see  §§113-114). 

After  the  conditions  (63)  are  satisfied,  the  solutions  of  (57)  are 

'  wt(e  ;  0  +P»(fl  +P?(fl,  (66) 


where  P?  is  linear  and  homogeneous  in  c|ue"v~'  and  c^e~w^,  and  where  P,a> 
is  homogeneous  of  the  third  degree  in  ^efv~11  and  c^e~a^'lt.  In  the  right 
member  of  (57)  the  coefficients  of  all  cosine  terms  are  real,  and  the  coeffi- 
cients of  all  sine  terms  are  purely  imaginary;  and  moreover  those  which  are 
multiplied  by  c|",  (c"')',  (OM"  differ  from  those  respectively  which  are 
multiplied  by  ci°,  (O1,  c|"(O'  only  in  the  sign  of  ^/F=\.  It  follows  that 
PI"  and  F?  have  these  properties  also.  The  u>,  and  wt  are  given  in  (53) 
and  (54). 

The  P[3)  and  P*  are  entirely  known  functions,  while  c,a)  and  cj"  are 
subject  to  the  relation 


248  PERIODIC    ORBITS. 

It  follows  from  the  properties  of  the  expressions  Pf}  (t)  and  P™  (t)  and 
c?=-c?  that  P|3)(0)=Pf(0)=0.  Hence  cf  and  cf  are  subject  to  the 
condition 

0.  (67) 


Therefore  but  one  undetermined  constant  remains  in  (66). 

The  divisors  introduced  at  this  step  can  be  found  as  they  were  in  the 
preceding  step.  They  are  the  determinant  of  the  fundamental  set  of 
solutions  of  the  z-equation,  j2  —  o>2,  andj2  — 9w2,  the  j-  —  or  appearing  with 
terms  which  involve  e*uV=It  ™*  jt,  and  the  /— 9w2  with  those  which  involve 
e-*ov=n  u*jt.  The  coefficients  of  ^  jt  contain  e'  as  a  factor. 

123.  Coefficients  of  X2. — The  differential  equations  at  this  step  are 

x"-2y(-[l+2A+GAecost+  •  •  -~k-  [~6 Seisin t  +  •  • 

(68) 
y"+2x't-[6Aesmt+  •  •  -]z4-  [l-A-3Aecost+  •  • 

where  X4  and  F4  are  of  even  degree  in  ewv/rri(  and  e~wV=It  considered  together. 
It  follows,  as  in  the  case  of  x2  and  y2 ,  that  in  the  right  member  of  the  first 
equation  the  coefficients  of  all  cosine  terms  are  real,  and  those  of  all  sine 
terms  are  purely  imaginary ;  the  opposite  is  true  in  the  right  member  of  the 
second  equation.  The  coefficients  of  c{uc{8)  and  (c"})4  differ  from  those  of 
c£u  c®  and  (c^)4  respectively  only  in  the  sign  of  V  —  i.  The  coefficients  of 
WyC*?)1  are  real>  independent  of  eu%/=T<  and  e~Wv/=7(,  and  involve  only  cosines. 
The  solutions  of  (68)  have  the  same  properties.  It  follows  from  these 
properties  and  cf  =  -of,  that  x( (0)  =  yt (0)  =  0. 

The  divisors  introduced  in  the  solution  are  the  determinant  of  the 
fundamental  set  of  solutions  of  the  x  and  ^-equations,  j2  —  16u>2,  j*  —  4co2, 
and  j  in  connection  with  the  terms  involving  e±4wv/rrr'gfn9  jt,  e*wv^~11  I™  jt, 
and  C8™jt  respectively.  The  coefficients  of  these  terms  contain  e'  as  a  factor. 

124.  Coefficient  of  X'A. — It  is  necessary  to  consider  this  step  in  order 
to  be  able  to  make  the  general  discussion.     The  differential  equation  defin- 
ing z6  is 

=  Z;>,  (69) 


where  it  is  found  from  (14)  and  properties  (a)  and  (d)  of  §  106  that  Z&  is 
made  up  of  real  cosine  series  multiplied  into  z3 ,  .T2 z3 ,  xt zl ,  x\z^,  y\ z, , 
ZjZ3,  z\,  and  of  real  sine  series  multiplied  into  y«z3  and  y4zlf  It  follows 
from  the  properties  of  zn  z3,  x2,  y2,  xt,  and  yt  that  Z5  is  of  odd  degree 
in  ew%/~'  and  e~^^'  taken  together;  that  the  coefficients  of  the  cosine  terms 


OSCILLATING   SATELLITES,   ELLIPTICAL  CASE. 

arc  real,  and  those  of  the  sine  terms  purely  imaginary;  that  cj"  ami  rj"  outer 
linearly;  that  the  coefficients  of  c*,  (ciu)'<f,  c\Hc?}c?>,  (O'CO1  differ  from 
the  coefficients  of  <f,  (OX",  c'^c?  and  (O'(O'  respectively  only  in 
tin  sign  of  V—  i;  and  that  the  terms  which  involve  cosjt  and  sin.;*  have 
e'  as  a  factor  in  their  coefficients. 

Now  consider  the  solution  of  (69).     The  discussion  proceeds  as  in  the 

of  (57),  and  it  is  found  that,  in  order  that  the  solution  shall  be  periodic, 

the  constant  terms  in    -  u>,  Z,  e-"*^1'  and  +«>,  Z»  ewv/^r<  must  be  zero.     It 

follows  from  the  properties  of  Z»  enumerated  above  that  these  conditions 

are  of  the  form 


where  At,  .  .  .  ,  A<  are  power  series  in  e.    Upon  reducing  by  means  of  the 
last  equation  of  (64),  we  get 


'      <f  +i      cf-0, 


(71) 


Since  cju  =  —  c|u  and  ciw=  —  c™,  these  equations  are  equivalent  and  define 
c™  provided 

_A,K     A,K* 
Al       L          U 

is  distinct  from  zero.  In  the  case  e  =  0  of  Chapter  VI  the  corresponding 
coefficient  was  distinct  from  zero.  Since  the  coefficient  in  the  present  case 
is  a  power  series  in  e  and  reduces  to  that  of  the  former  case  for  e  =  0,  it 
follows  that  it  is  distinct  from  zero  for  e  sufficiently  small. 

When  equations  (71)  are  satisfied,  the  solution  of  (69)  has  the  form 

Zs  =  cu)eUv^T,u,i+CiU)e-w-,,u,|  +  p«.(0+pa,W+pt.»W>  (72) 

where  P"(0  is  linear  and  homogeneous  in  c^'e^^7'  and  cJ"e~w%ArT/  taken 
together,  and  P"'(0  and  PJ"(0  are  homogeneous  of  the  third  and  fifth 
degrees  respectively  in  e"*^1'  and  e~a^~'  taken  together. 

It  follows  from  the  facts  noted  above  that  Zt  is  unchanged  if  c|", 
c|",  +  V—i  are  interchanged  with  cj",  cj",  —  V  —  1,  and  that  the  coeffi- 
cients of  the  cosine  terms  are  real;  hence  the  solution  (72)  has  the  same 
properties.  Consequently,  since  cj"  =  —  c"},  c,u>  —  —  c,*,  it  follows  that 
WO)  =  P,U)(0)  =  P*(0)  =  0;  and  since  z,  (0)  =  0,  that 

O.  (73) 


250  PERIODIC   ORBITS. 

125.  The  General  Step  for  the  x  and  ^-Equations.  —  Suppose  z,,  xt, 
j/,,  .  .  .  ,  z^-i,  z2(,-2,  2/2X-2  have  been  computed  and  that  they  have  the 
following  properties: 

(A)  The  x2J  and  y2}  are  even  functions  of  eWv/^T<  and  e'"'*^1'  taken  together, 
and  the  z2j+i  are  odd  functions  of  eav^7t  and  e~uv~'  taken  together. 

(B)  In  the  x2J  and  z2;+i  the  constant  parts  of  the  coefficients  of  all 
cosine  terms  are  real,  and  those  of  all  sine  terms  are  purely  imaginary. 

(c)  In  the  y2,  the  constant  parts  of  the  coefficients  of  all  cosine  terms 
are  purely  imaginary,  and  those  of  all  sine  terms  are  real. 

(D)  In  the  x2J  and  y-is  the  highest  powers  of  euV~'  and  e""^^1'  are  2j, 
and  in  z«+i  they  are  2j+\. 

(B)  The  coefficients  of  (<f)*(Ov  '  '  '  (<a°)'l(Ot'(<f)*  '  '  '  (<•?*)*-  in  the 
expressions  for  x2},  y2),  and  z2i+l  differ  from  the  coefficients  of 
(O*(O*  '  '  '  WWTCclT  •  •  •  (cT)*"  only  in  the  sign  of  -/=T. 

(F)  The  constants  of  integration  arising  at  the   step  2j  -f  1,   viz., 

cfj+1)  and  c"+I  (j=l,  .  .  .  ,  v  —  2),  must  satisfy  the  relations 


where  M"^1'  is  a  power  series  in  e,  in  order  that  the  solution  at 
the  step  2.7+3  shall  be  periodic.  The  constants  cf  ~u  and  c"""" 
remain  arbitrary  at  the  step  2v. 

(G)  The  divisors  introduced  at  the  step  2j  are  the  determinant  of 
the  fundamental  set  of  solutions  of  the  x  and  ^-equations  and 
k,  A2-4co2,  &2-16u2,  .  .  .  ,  A;2-4jV  in  the  coefficients  of  <?*kt; 
and  at  the  step  2j+l  they  are  the  determinant  of  the  funda- 
mental set  of  solutions  of  the  z-  equation  and  k,  &2—  o>2  A;2—  9<o2, 


The  differential  equations  which  define  x2v  and  y2v  are 


xi-     [GAesint  •  •  -]x2v-[l-A-  ~  * 


2 


It  follows  from  the  properties  (a),  (b),  (c),  and  (d)  of  §106  that  X 
and  Y2V  are  even  functions  of  zu  .  .  .  ,  z2>._i  taken  together;  that  in  X 
the  coefficients  of  all  terms  which  are  of  even  degree  in  y2,  yt,  .  ,  .  ,  y}v_ 
taken  together  are  sums  of  cosines  of  integral  multiples  of  t;  that  in  Xtv 
the  coefficients  of  all  terms  which  are  of  odd  degree  in  t/2,  yt,  .  .  .  ,  ytv_t 
taken  together  are  sums  of  sines  of  integral  multiples  of  t;  that  the  last  two 
properties  are  reversed  in  the  case  of  Ytv;  and  that  if  in  Xtv  or  Ytt  the 
general  term  has  as  a  factor 


A  .      .  -A 


then 

2X.+ 


OSCILLATING    SATELLITES,  ELLIPTICAL   CA  251 

It  follows  fi om  these  properties  and  (A),  .  .  .  ,  (<;)  that  X^  and  }',„  are 
even  functions  of  e"^7'  and  e~"v~"  taken  together;  that  in  A'u  the  constant 
parts  of  the  coefficients  of  all  cosine  terms  are  real,  and  those  of  all  sine 
terms  are  purely  imaginary;  that  in  !'„  the  last  property  is  reversed;  that 
in  A'j,  and  }',„  the  highest  powers  of  e"v~'  and  e-«*v'=71  are  2v;  and  that  the 
coefficients  of  (O^O*  •  :  •  (OHO^c*)*'  •  •  •  (ef1)*-  differ  from  the 
coefficients  of  (c,ft))*(O*  •  •  •  (c1a>)''(clU))*'(O*f  •  •  •  (c**)'-  only  in  the  sign 
of  V—l. 

It  follows  from  the  form  of  (74)  and  these  properties  that  the  periodic 
solutions  of  equations  (74)  (i.  e.,  the  particular  integrals)  have  the 
properties  (A),  .  .  .  ,  (o)  so  far  as  they  pertain  to  x^,  and  ytj;  and  then 
from  c,U)=  -ci",  that  x^(0)  =  yu(0)  =  0. 

126.  The  General  Step  for  the  ^-Equation. — The  differential  equation 
defining  zjr  M  is 

zl+i+[A+3Aecost+  •  •  •  ]  z,r+l  =  Ztf+l .  (75) 

It  follows  from  the  properties  (a),  .  .  .  ,  (g)  of  §106  that  Zt,+i  is  of 
odd  degree  in  z,,  z,,  .  .  .  ,  z»-i  taken  together,  and  that  it  contains  z,,_, 
linearly.  In  Ztf+l  the  coefficients  of  terms  which  are  of  even  degree  in 
t/,.  .  .  .  ,  //:„  taken  together  are  cosines  of  integral  multiples  of  /,  and  the 
coefficients  of  those  terms  which  are  odd  in  the  same  quantities  are  sines 
of  integral  multiples  of  t. 

It  follows  from  these  properties  and  (A),  .  .  .  ,(G)  of  §125  that  Ztf+i 
is  an  odd  function  of  e"*^1'  and  e~~</=lt  taken  together;  that  the  highest 
power  of  e"v^ii  an(j  e-«v=r<  js  2i»+l;  that  c,8*'"  and  cj*"l)  enter  linearly; 
that  the  constant  parts  of  the  coefficients  of  all  cosine  terms  are  real, 
and  that  those  of  all  sine  terms  are  purely  imaginary;  and  that  the  coeffi- 
cients of  (c!")'1  •  •  •  (O'1  (O*1  •  •  •  (cj")'-  differ  from  the  coefficients  of 
((-i11)'1  •  •  •  <£*)*  (ci")*1  •  •  •  (O*"  only  in  the  sign  of  V=\. 

Now  consider  the  solution  of  (75).  The  conditions  that  it  shall  be 
periodic  with  the  period  T  are  that  in  —  u>tZ,r+I  and  +wlZtr+i  the  constant 
parts  of  the  coefficients  of  ^/=lt  and  e'""^1'  respectively  shall  be  equal  to 
zero.  The  zjr_i  enters  Z,,+,  linearly  and  has  the  same  coefficient  that  z,  has 
in  Z, .  Hence,  from  the  relations  of  the  preceding  paragraph  and  equations 
(70),  we  have 

I.I.J-MI-MAJ    ^an.^-^p       ,„<„    ,^,ft  ^ 

where  P,r+,  is  a  polynomial  of  odd  degree  in  c"}  and  cj1'  taken  together.  It 
is  supposed  that  c,a/+l)  and  c?'+"  0  =  1»  •  •  •  >  "~ 2)  have  been  eliminated 
at  the  successive  steps  by  the  equations  corresponding  to  (71).  If  the 
general  term  in  P*^  is  (c"'X(O*,  then  j  and  k  satisfy  the  relation 

(77) 


252  PERIODIC    ORBITS. 

On  reducing  (76)  by  means  of  (64),  making  use  of  (77)  and  (F)  of  §125, 
and  the  fact  that  c^1>=  —  c[l\  it  is  seen  that  equations  (76)  are  equivalent, 
and  that  cf"~ l)  =  cf"~  '  is  uniquely  determined  as  a  power  series  in  e.  Since 
in  Z2v+l  the  coefficient  of  (O'1  •  •  •  (c<")'<  (O*1  •  •  •  (cf)*-  differs  from  that 
of  (O'1  '  '  '  (OHO*1  '  '  '  (c^)*"1  only  in  the  sign  of  V  —  1,  it  follows  that 
the  solution  has  the  same  property,  and  from  this  that  z2H-i(0)  =0. 

For  small  values  of  v  there  will  be  no  other  terms  than  those  considered 
above  in  —  w^Z2v^\  and  -\-wlZ2i>+i)  which  are  constants  multiplied  by  the 
exponentials  euV=*'  and  e-wv/=I<  respectively.  When  there  are  no  other  terms 
and  when  equations  (76)  are  satisfied,  the  solution  of  (75)  is  periodic  and  z2t,+1 
has  all  the  properties  of  z2J_i  specified  in  §  125. 

But  since  u=N/n,  where  n  and  N  integers,  there  is  a  value  of  v  for  which 
other  terms  of  the  type  in  question  can  arise.  It  follows  from  the  properties 
of  Zw+i  that  — m,Z2x+i  and  +^1Z2i/+i  contain  respectively  the  terms 

•  •  •  +atcoskt-\-  •  •  • 

•  —V  —  lbtsmkt—   •  •  •], 

\  (78) 

•  •  -{-atcoskt+  •  •  • 


Now  e<*»+«<''V=i<  =  e"v:=*t[cos2vut  +  V—  ism2vut],  Consequently  terms 
of  the  type  in  question  will  arise  if  Ivu  =  k,  k  being  an  integer.  Upon 
substituting  the  value  of  co  ,  this  relation  becomes  2  m  =  kN,  which  can 
be  satisfied  when  2i>  becomes  a  multiple  of  N.  Suppose  the  integer  n  is 
odd.  In  this  case  when  2v  —  N,  the  smallest  k  satisfying  the  relation, 
viz.  k  =  n,  is  obtained.  The  term  in  which  this  occurs  is  multiplied  by 
X2"+1/1  e"  =  \N+l/>  en.  But  if  n  is  even  and  N  odd,  the  relation  is  satisfied  first 
for  increasing  values  of  v  when  2v  =  2N,  and  then  k  =  In.  The  term  in  which 
this  relation  occurs  is  multiplied  by  X2"+'A  e2"  =  X2Ar+'A  e2".  After  terms  of  this 
type  once  appear  they  in  general  occur  similarly  at  all  subsequent  steps  of 
the  integration. 

The  coefficients  of  eWv/Tri(  and  e~a^=^'  obtained  from  (78)  are  respec- 
tively -C2x+i(O2*+1  and  +C*2^+i(c21))2"+1,  where  C2v+i  is  a  constant  multiplied 
by  e"  or  e2"  according  as  N  is  even  or  odd.  Therefore,  when  these  terms 
arise  we  have  in  place  of  equations  (76) 


Consequently  cj2"""  and  Cj2"""  are  determined  in  this  case  as  well  as  in  that 
in  which  the  terms  multiplied  by  Ctv+l  do  not  arise.  This  completes  the 
proof  of  the  possibility  of  constructing  the  solutions. 


OSCILLATING    SATELLM  !•>.    KI.L1PT1CAL   CASE.  253 

APPLICATION  OF  THE  INTEGRAL. 
127.  Form  of  the  Integral.     Kquations  (1)  can  be  written  in  the  form 

<?A-2—  =  —>  <f  T  =  dU 

(80) 


After  the  transformations  (10)  and  (11),  these  equations  have  the  form 


"^z" 

wliere  now  U  is  a  power  series  in  x,  y,  and  z*  and  contains  no  terms  lower 
than  tin  >eeond  degree  in  x,  y,  and  z.  It  contains  terms  independent  of 
X  and  others,  for  the  particular  transformation  (11),  multiplied  by  X  to  the 
first  degree  only.  The  coefficients  in  the  series  for  (7  are  power  series  in 
c  whose  coefficients,  in  turn,  arc  periodic  in  t  with  the  period  2r,  and  which 
reduce  to  constants  for  t  =  0. 

The  first  terms  of  U  are  seen  from  equations  (10)  and  (11)  to  be 

•  -~\xy 


£*»!  +  ,wi  -  3  (  «,  -   a 

LT,  7,  V|  't 


i-  r»ri  +  3  (  4,),  -r(b.i)ecosf+ 
'  t  \'  i          'i    / 


I 


(82) 


The  integral  of  equations  (81),  analogous  to  the  Jacobi  integral  in  the 
case  where  U  does  not  involve  t  explicitly,  is 


dt+C,  (83) 


where  (dU/df)  is  the  partial  derivative  of  U  with  respect  to  /  so  far  as  t  occurs 
explicitly,  and  not  as  it  enters  through  x,  y,  and  z.  This  partial  derivative 
is  zero  for  e  equal  to  zero,  and  therefore  it  contains  e  as  a  factor. 


254  PERIODIC    ORBITS. 

128.  The  Integral  in  Case  of  the  Periodic  Solutions.  —  In  the  periodic 
solutions  which  have  been  considered,  x,  x,  y,  and  y'  are  expansible  as 
power  series  in  X,  while  z  and  z'  are  expansible  in  odd  powers  of  A*.  It 
follows  from  property  (a)  of  §106  that  U  is  a  power  series  in  z\  Therefore 
when  the  expansions  of  all  these  variables  are  substituted  in  (83),  the  result 
is  a  power  series  in  integral  powers  of  X  of  the  form 

F  =  Fl\+Ft\*+  •  •  •  +FV\*+  •  •  •   =<?,  (84) 

where,  of  course,  F  and  the  F}  involve  the  integral  sign  which  arises  from 
the  right  members  of  (83),  and  where  C  is  the  constant  of  integration. 

Since  (84)  converges  and  is  satisfied  for  all  |X|  sufficiently  small,  it 
follows  that 

FV  =  CV  (v=l,  ...»),  (85) 

the  Cv  being  constants.     Since  the  series  for  x,  y,  and  z  have  the  forms 


it  follows  that  f\  involves  x2,  x'2,  y^,  y'2,  .  .  .  ,  x2v_2,  o4_2,  y2v_2>  y'2v_2, 
zl,z{,  .  .  .  ,  z2,-i,  z»-i«  Equation  (85)  therefore  has  the  form 

/r)O 
jw  JWMJ  y^n  Z2j+i>  t)dt  =  Cv)        (86) 

where  Pv  and  Qv  are  polynomials  in  the  indicated  arguments  with  t  entering 
the  coefficients  in  sines  and  cosines.  The  subscript  j  runs  from  0  to  v—  1. 
The  derivatives  enter  (86)  only  in  the  form  x^^x'^.^  ,  y't)-ty'tv-ti,  and 
z«-i22K-2j  •  Suppose  the  general  term  of  Pv  or  Qr  is 

x,  x        x;  X',      x"  x",, 

x      •••x"*?y       •••jy*'2         *  *   *  z  " 

2M,  2^       J2p.\  V^,          2^+1  2^',,+l   ' 

The  exponents  and  subscripts  satisfy  the  relation 


The  partial  derivative  of  Qn  with  respect  to  t  is  taken  only  so  far  as  t 
enters  explicitly  in  the  coefficients  of  xzj  ,  .  .  .  ,  zlj+1,  but  the  integral  must 
be  computed  for  t  entering  both  explicitly  and  also  implicitly  through 


the  xt} 


It  was  shown  in  §125  that  the  xzj,  the  y2J!  and  the  zv+1  have  the  form 


~     —     y    ~<2*)/,2ito-v/=n'(        .,     _     y    7/<2*>«2tu\/=T<       -         _       y 

xtj—    z,  xtj  e          ,    ytj—    LI  ytj  e          ,    ztj+1—     z, 
t--j  »—  j  t—j 


MX  II, LATIN".    -AIKLI.II  K>.    ELLIPTICAL    CASE. 


255 


where  the  j^\  the  //£",  and  the  2,%"  are  power  series  in  <  whose  coefficients 
are  periodic  with  the  period  2r.  In  j£"  and  2,7+V  the  eorHieients  of  the  cosine 
terms  are  real,  and  those  of  the  sine  terms  are  purely  imaginary :  the  opposite 
is  true  in  y{"\  It  follows  from  these  projKTties  and  those  of  Ff  enumerated 
on  page  254  that  Pr  and  dQf/dt  can  \w  written  in  the  form 


/',-    2 


*=    2 


(88) 


where  P"'  and  /C'  are  periodic  with  the  period  2r.  Since  in  U  the  coefficient  > 
of  odd  j>owers  of  y  are  multiplied  by  sine  series,  it  follows  from  the  properties  of 
the  x™},  the  y™\  and  the  z("£"  that  in  P"}  the  coefficients  of  the  cosine  terms 
are  real,  and  those  of  the  sine  terms  are  purely  imaginary;  the  opposite  is 
true  in  R*\ 

The  integrals  coming  from  j  -*•'  dt  are  of  the  types 


.'  ../ 


Binjt> 


e- 


Therefore  we  have 


(89) 


C.K)) 


where  the  S™  are  i>eriodic  with  the  period  2*.  Moreover,  the  coefficients  of 
the  cosine  terms  are  real,  and  those  of  the  sine  terms  arc  purely  imaginary. 
It  follows  from  these  properties  that  (85)  can  be  written  in  the  form 


F,=   2 


L» 


(91) 


where  the  f'"'  are  j)eriodic  with  the  period  2*-.     The  coefficients  of  the  cosine 
terms  are  real,  and  those  of  the  sine  terms  are  purely  imaginary.     If  we  let 

<**"=i*  =  ff>  (92) 

and  make  use  of  the  fact  that  the  F"'  are  periodic  with  the  period  2*,  we 
get  from  (91) 


2  / 

«--r 


2  I 

tm-f 


2  1 

lm—f 


These  equations  and  (91)  can  be  satisfied  only  if  either 

F,W-C,         FJ"  =  0  (k--, -l.+l 

or 


TI  j»_/~i      fai} 
o\  -or.   (.yo; 


(94) 


1     , 
<r_,  , 


<r_, 


1, 
1, 
I, 


1 


1 


?r,  .   .   .   ,  a!!, ,     1,      a,  ,  .  .   .  ,  <rr 


=  0. 


256  PERIODIC   ORBITS. 

This  determinant  is  the  well-known  product  of  the  differences  of  the  a±, 
taken  in  all  possible  pairs,  and  is  distinct  from  zero  unless  a  relation  of  the 
form  a,  =  as  is  satisfied.  Since  a,  =  a\  ,  dj  =  a\  ,  this  relation  can  be  satisfied 
only  if  a\~1=l;  or,  because  of  (92),  only  if 


Since  u  =  N/n,  this  equation  can  be  fulfilled  only  if  &(i—j)/n  is  an  integer. 
But  then  two  or  more  of  the  exponentials  of  (91)  are  equal  in  value,  and 
the  number  of  terms  under  the  summation  sign  is  reduced  by  combining 
similar  ones.  With  this  understanding  as  to  the  reduction  of  (91),  A  can 
not  vanish  and  equations  (94)  must  be  fulfilled.  It  is  clear  that  when  these 
equations  are  satisfied  all  relations  of  the  form  of  (93)  are  satisfied. 

The  F™  are  explicit  power  series  in  e,  and  they  also  involve  e  implicitly 
in  w,  which  is  a  power  series  in  this  same  parameter.  Now  w  enters  in  two 
ways.  It  is  introduced  as  a  factor  of  certain  terms  either  to  the  first  or 
second  degree  by  the  derivatives  which  occur  in  (83),  and  to  the  first  degree 
by  the  integral,  as  is  shown  by  (89).  The  integral  also  introduces  it  in  the 
denominators  in  the  form  f—  4&W.  We  shall  substitute  for  w  its  series 
in  e  wherever  it  enters  in  the  first  way.  This  will  not  change  the  character 
of  the  convergence.  But  where  w  enters  in  the  second  way  we  shall  regard 
e  as  an  independent  parameter  and  leave  it  implicitly  in  w.  Then  equa- 
tions (94)  can  be  written  in  the  form 

Fi°>  =  ^F^e]=Cv=  i  Cv.,e>, 

J**0  *=0 

F»  =  S  F»  e'=0         (k=-v,  ...,-!,+!,...,  +„). 

J=0 

Since  these  equations  are  identically  satisfied  in  e,  we  have 

F<0)  =  C  F(l>  =  0  (95^) 

1  y.J        ^v,)1  L   v,S  —  V.  \.V<JJ 

The  F™}  are  sines  and  cosines  of  integral  multiples  of  t.  On  making  use 
of  the  properties  established  in  §  126,  we  have 


F™  =       [a™,,,  cosp«+  v^ 

J>=0 

F*  =  S  [<),„  cos  pt  +  V^-T  &£.„  sin  pt]  =  0. 

P=0 

Since  these  equations  are  identities  in  t}  we  have  finally 


(9b) 


"-   II.I.VI1M,    >\IKI.I.IIKS.    KU.II'TICAL    CA-I  257 

129.  Determination  of  the  Coefficients  of  z,,^  when  r  =0.  —  Equations 
(96)  are  relations  anionn  the  coefficients  of  the  solutions  and  may  be  used  for 
checking  tho  computations.  The  control  is  very  effective  because  at  each 
step  all  tho  preceding  coefficients  are  in  general  involved.  But  equations 
(96)  can  also  be  used,  step  by  step,  for  the  determination  of  the  coefficients 
of  the  expansion  for  z  when  the  coefficients  of  the  expansions  for  x  and  y 
are  determined,  alternately  with  those  for  z,  from  the  first  two  equations  of 
(13).  Before  taking  up  the  general  problem  we  shall  treat  the  case  of  e  =  0. 
\\hcn  this  condition  is  >ati.-fied  the  integral  becomes 


(97) 

£" o  —      _»)»     +  _Wi  ' 

'i  'i 

In  this  case  w  =  \fA  and  equations  (87)  become 

+>  +/  +/ 

Zjj  =  —  o^t  e^  ,  yt)  =  2  pti  e^>      ~  ,  ztl^.l  =     2 

where  the  o,a/',  /3^',  and  •y^+l>  are   constants.     We  shall  show  how  to 
compute  the  7£*+i"  for  successive  values  of  j. 

Terms  for  j  =  Q.     In  this  case,  since  x0=sy0  =  0,  equation  (97)  becomes, 
as  a  consequence  of  the  last  of  (98), 


Since  this  equation  is  an  identity  in  t,  we  get  4Ay[~"  y\"  =  (\.  Since  C,  is 
unknown,  this  equation  imposes  no  relation  on  y[~l)  and  -y"'.  But  since 
2,  (0)  =  0,  it  follows  that  y[~"  =  —  7"',  and  there  remains  the  single  unde- 
termined constant  y™. 

Terms  in  x  and  y  for  j=l.     It  follows  from  (13)  and  (14)  that  x,  and 
yt  are  defined  by  the  equations 


xZ-2y',-(l+2A}x,=  \Bz\=  \  -,     (flg) 


The  solutions  of  these  equations,  which  have  the  period  2r/\/A)  are 

,  .  3B(7;")i 

2(1  - 


(100) 


258  PERIODIC    ORBITS. 

Terms  in  z  for  j=l.     It  follows  from  (97)  that  the  terms  for  j=l  are 


Since  z,  has  the  form 


equation  (101)  gives  rise  to  the  relations 

Y=-^Ayryr, 


There  is  another  equation  which  is  useless  for  present  purposes  because  it 
involves  the  unknown  constant  (72  .     Since  z3  (0)  =  0,  we  have  also 


.  (104) 

It  follows  from  equations  (103)  and  (104)  that 

yi-*=-y?,       73-l)=-731).  (105) 

In  order  that  7j~3)  shall  be  the  same  as  determined  by  both  equations  of 
(103),  we  must  impose  the  condition 


(I,V  ,   ftK,_n 

)    ^3-(7.)      0, 


which  determines  7"'  except  as  to  sign.     Then  73~3)  and  73"  are  uniquely 
determined  in  terms  of  7"',  but  73"  remains  so  far  arbitrary. 

The  problem  for  e  =  0  was  treated  in  Chapter  VI.  If  we  compare 
equations  (100)  of  the  present  work  with  equations  (42)  of  page  211,  we 
find  that  7ilJ  and  ct  are  related  by  the  equation 


Upon  making  use  of  this  relation,  it  is  seen  that  equations  (106)  of  this 
chapter  and  (44)  of  Chapter  VI  are  identical. 

Terms  in  x  and  y  for  j  =  2.     It  follows  from  (13)  and  (14)  that  we  have 
in  this  case 

/Y," O  «/  *  _    f  1     I    O    A  1  /*•    —  Q  Z?  ty    /y      I     P  a  §     .1  ,  O  /)•    _  _  /a  1  /7/    —  f)  I  1  07  1 

•vi  '~~  *  i/4  ^^  I  J-     1^  ^  -^-1  ^d  —  *-*  •*-*  ^l  *3    1^       4  J  c/4  ^^          ^         L  ^T- J  t/4  ~~"  V^j^  ^  ^  XVI I  j 

where  P4  and  Q4  are  entirely  known  periodic  functions  of  t  having  the  period 
T.     We  wish  the  details  of  the  solutions  of  these  equations  only  so 


OSCILLATING    8ATELLITKS,    KLLIPTICAL   C.\ 


259 


far  a*  they  depend   upon   the  undetermined  constant  >i"=—  y't~l\     So  far 
as   these   terms    arc   involved,   the   right    member   of   the   first  equation    is 

•'*-"].     Hence  the  solutions  of  (107)  arc 


(108) 


U    =   -f 


/=7ll  _l_  7J 

Vi. 


where  ~f\  and  ^>4  are  known  periodic  functions  of  /. 

'(N  in  z  forj  =  2.     It  follows  from  (97)  that  these  terms  are  defined  by 

2z(z't+2Azlzk=-2x',i'<-2y'1y'<+2(\+2A)xtx<+2(l-A)yty< 

(109) 


where  /j1,  is  a  known  periodic  function  of  t.     The  expression  for  z,  has  the 
form 


(110) 


On  substituting  this  expression  in  (109)  and  equating  the  coefficients  of  the 
several  powers  of  «VJV=T',  beginning  with  e~'>:ivrrT',  we  get 


4  y\"ylr"  =  known  function  of  7!"  =-8  A 


of 


[  "/^ 


There  is  also  an  equation,  coming  from  the  terms  independent  of  e^*^1', 
which  involves  the  unknown  T4and  need  not  be  written.  The  first  equation 
uniquely  defines  7j~B,  which  equals  —  7,"'.  In  order  that  the  second  and 
third  equations  shall  be  consistent,  we  must  impose  the  relation 


?-7A  +  UA*  +  C«]W)'7;"  =  known  function  of  y[». 


(112) 


Therefore  7!"  is  uniquely  determined,  since  its  coefficient  in  this  equation  is 
positive;  and  then  the  second  or  third  of  (111)  defines  7}"",  which  is  the 
negative  of  7,"'. 

Now,  on  imposing  the  condition  that  z,(0)=0,  we  get  7j~"=  —  y'",  and 
7"'  remains  undetermined. 

All  succeeding  steps  are  precisely  similar  to  the  one  which  has  just 
been  explained.  The  parts  of  the  equations  which  contain  undetermined 
coefficients  differ  from  those  of  (111)  and  (112)  only  in  the  subscripts. 


260  PERIODIC    ORBITS. 

130.  Case  when  e  ^  0.  General  Equations  for  zt  .  —  It  will  now  be  shown 
that  when  the  coefficients  of  the  series  for  x  and  y  are  determined  from  equa- 
tions (13)  and  (14),  the  coefficients  of  the  expansion  for  z  can  be  determined 
from  the  integral  (83)  .  We  shall  need  the  partial  derivative  of  U  with  respect 
to  t  so  far  as  this  variable  occurs  explicitly.  We  find  from  (82)  that 


•  •  -~\xy  + 

Jz2  +  [jj(--<b,3  -  psrs 
[6  (r<6)3  -r  (ov^e  cos  £+  •  •  •  Jz?/A-[!  (  o»i-p»)  e  sin  t+  •  •  -J?/X 


(113) 


Since  z0  =  2/0  =  0,  we  find  from  (82),  (83),  and  (113)  that  the  integral  is 

z'*  =  —  [A+3Aecost+  •  •  -]z2  — f[3Aesmt+  •  •  -]z\dt-\-Ci.      (114) 
In  the  notation  of  equations  (87),  the  expression  for  z±  has  the  form 

where 


-    _   y  z-  -     _        2 

i        ~  "  zi.t    K  i  z\    '  ~   "  zi.  t 

t-o  t-o 

-" 


z<-«_   y 
n,i       * 

p=0 


(115) 


On  substituting  these  expressions  in  equation  (114),  it  is  found  that 


m 

[A+3Aecost+ 
sint  + 


OSCILLATING    SATELLITES,    ELLIPTICAL   CASE.  261 

Before  the  integration  the  series  for  z[~"  and  z|"  must  be  substituted 
from  (115).  Consider  the  coefficient  of  e'*^^1'  under  the  integral  sign. 
It  is  a  sum  of  cosines  ami  sines  of  integral  multiples  of  t.  Suppose  V^lAi 
and  B,  are  the  coefficients  of  cosjt  and  sinjt  respectively.  It  follows  from 
(89)  that  in  the  integral  we  have  in  place  of  these  terms 


respectively.  The  corresponding  formulas  for  the  coefficient  of  e**v=T« 
are  obtained  from  these  simply  by  changing  the  sign  of  co.  The  terms 
independent  of  the  exponentials  which  involve  the  cosine  and  sine  of  jt 
are  divided  by  +j  and  —j  respectively.  Consequently,  in  all  cases  it  is 
easy  to  write  down  the  explicit  equations  for  the  identities. 

In  the  notation  of  (91)  the  coefficients  of  e-*^/=T',  e+ta*  ~',  and  e°  in 
(116)  are  respectively  F[~",  F?,  and  F?,  and  we  have 


Since  these  functions  are  power  series  in  e,  we  have  in  accordance  with  the 
notation  of  (95) 

fV-FS-O,  FZJ-CW.  (118) 

And  these  functions  in  turn  have  the  form 


a«  ,cospH-  v^T6«  f  si 


Since  these  equations  are  aU  identities  in  /.  we  have 

•  n=n  i\ 

(119) 


In  order  to  get  the  explicit  values  of  these  constants  in  terms  of  the 
*!.*!i,  K,  <!,.  and  0JIJ.,,  we  must  refer  to  equation  (116).  We  find 
from  (115)  that 


«J!S 


=0 


262  PERIODIC    ORBITS. 

whence 


0    +«ku«1'.(fue 

0+0  +(21'.(r1))2>2+---, 


0      +z 
0+0 


zi-»z[m  =0+0 

-"C   =     0     +z 

z«>2l"-'>  =  o  +« 


(120) 


Equating  to  zero  the  coefficients  of  the  various  exponentials  of  (116),  we  get 


3Aesmt+ 


-J[ 


r 
-  \[3Aesint+ 


where  the  coefficients  under  the  integral  sign  /  must  be  transformed  by 
equations  (117)  instead  of  forming  the  ordinary  integrals,  and  where  (7t  is  an 
undetermined  constant.  These  equations  are  power  series  in  e,  and  setting 
their  coefficients  equal  to  zero  we  have  equations  (119). 

131.  Coefficients  of  e\  —  On  referring  to  (120),  we  find  for  these  terms 


,._  _  - 

—  Uolt0      =  —      .LQ  )  ,      —01:fl    -   —         l<9    ,  01>0     li0— 

Since  2,(0)  =0,  we  must  add  to  these  equations  z{^l)  +  2j"0  =  0.     It  follows 
from  these  equations  that 

ul  =  A,  2,(ro"=-C,  4A(C)«  =  C-llQ,  (122) 

and  z"o  =  ai!o,o  remains  as  yet  undetermined  since  C",,0  is  an  unknown  constant. 


OSCILLATING    SATELLITES,    ELLIPTICAL   CASE. 


263 


132.  Coefficients  of  e.—  On  referring  to  (120),  (117),  and  (115),  we  get 
at  this  step 


r«j-»  [  _«;-»  sin<+  v^Ttf:,!',  cos<]  =  -3(2;:e") 


0,i 


Jr",  CGS/+  v^T/S^ 
cos*]  -«{».[  -a^si 


-2«0  v^ 
T^:,",  cos<]} 


=lft?t  sini]  J 


(123) 


Since  these  equations  are  identities  in  t,  we  find,  after  making  use  of  equa- 

tions (122),  that 


0Ilfl 


. 

t     , 


1.1 ' 


+2  A 


i.T.V  -  6  A  (f«)', 


264  PERIODIC    ORBITS. 

From  the  first  two  sets  of  these  equations  we  get 


_  4    2 


«> 


The  first  of  the  last  three  equations  imposes  no  condition  upon  the  unknown 
coefficients  since  it  involves  the  undetermined  Clfl,  and  the  second  and 
third  equations  of  the  last  set  become  identities.  The  coefficients  a^,"  and 
°u,o>  which  are  still  undetermined,  are  not  involved  in  these  equations. 

The  fact  that  «,  is  zero  was  known  in  advance,  for  it  was  proved  in 
§  120  that  w  is  a  series  in  even  powers  of  e.  It  has  also  been  shown  that 
z{~l)  and  z",  aside  from  constant  factors,  differ  only  in  the  sign  of  V—i. 
Since  2(0)  =0  these  constant  factors  differ  only  in  sign,  from  which  it  follows 
that  a^^  differs  from  a["Jip  only  in  sign,  while  /3^,lj,  and  $",„  are  equal  for  all 
j  and  p.  Applying  the  condition  z(0)  =  0  to  the  terms  under  consideration  at 
present,  we  have  z£u(0)  +  z"}  (0)  =  0.  On  making  use  of  (124),  this  equation 
leads  to  the  result 

o£lS  =-«&,,  (125) 

and  a{"t0  alone  remains  undetermined  at  this  step.  Of  course,  it  should  be 
noted  that  aj"  fl  and  /3{"  ,t  are  expressed  linearly  in  terms  of  the  undetermined 
constant  z"i  =  a"o,0,  whose  value  will  be  fixed  when  we  treat  the  coefficient 
of  X2  in  the  integral. 

133.  Coefficients  of  e-  and  e*.  —  From  the  consideration  of  this  step  we 
can  infer  the  character  of  the  process  in  general.  Because  of  the  relations 
between  a^,",  p[~"f  and  a{"t,,  $",„  it  is  sufficient  to  equate  to  zero  the 
coefficient  of  &°>^*'  in  (116).  Since  we  are  interested  only  in  the  possi- 
bility of  determining  the  unknown  coefficients,  it  will  be  sufficient  to  write 
out  the  equations  explicitly  only  so  far  as  they  involve  these  unknowns. 
Upon  equating  to  zero  the  terms  independent  of  t  and  the  coefficients  of 
cos<,  cos2<,  sint,  and  sin2i  in  order  in  the  coefficient  of  e2,  we  find 


=o          - 

Pi,2,i        J/1,2,U  -'l.ol,I,Z  ),2,*>  /•.  no\ 

(1)  n.(1)    =  f  co>          -  4  \/~A  za)  a11'    —  f  <0> 

yi  ali2il  —  j  Ii2il  ,  ±  v  A  zli0  a1>2  2  —  7  1,2,2  ,  J 


where  /0)  is  known  and  where  {/{",,  ,  .  .  .  ,  /[°22  are  homogeneous  functions  of 
the  second  degree  in  z("0  =  a"o,0  and  aj"  „  ,  and  are  linear  in  a"J  0  alone.  The  first 
equation  uniquely  determines  w2  ;  the  remainder  determine  /3"2  ,  ,  .  .  .  ,  a"2,2 
uniquely  when  z™0  and  a['|  „  become  known,  as  they  do  when  the  coefficient 
of  X2  is  considered.  The  coefficient  a"2.0  is  so  far  entirely  arbitrary. 

The  coefficients  of  ek  lead  to  similar  equations.  The  first  has  cot  in 
place  of  co2,  and  the  left  members  of  the  remainder  involve  j8"i,,,  .  .  .  , 
$",»,  o"i.!,  .  .  .  ,  a"i,t,  the  numerical  coefficient  of  j8"iip  and  a"i,p  being 
—  2p\/JzJ'o  .  The  right  members  are  homogeneous  second-degree  functions 
of  a"o,0,  .  .  .  ,  a^_li0.  The  a"i,0  remains  arbitrary. 


OSCILLATING    SATELLITES,    ELLIPTICAL   CASE.  265 

134.  General  Equations  for  v=  1. — The  coefficients  of  z,  are  determined 
from  Ft  =  0.     From  (82),  (83),  and  (113)  we  find  explicitly  that 

2z'lz'3+2[A+3Aecost+  •  •  •]zlz>=-(x't)t-(y',)'+[l+2A+GAecost+  ••• 


*!+2j[ 


3Aea\nl+ 


-2 
-2 


j  [ 
J  [34esin<  +  •  •  •]zlz,dt+  f  [3(pi-p 


(127) 


The  xt  and  j/,  are  determined  from  equations  (55),  and  it  is  seen  from 
these  equations  that  they  are  homogeneous  of  the  second  degree  in  the 
coefficients  of  z,.  Other  properties  of  the  solutions  are  given  in  §121, 
among  which  is  that  they  are  of  even  degree  in  e"N/=T<  and  e~aV=ril. 

The  expression  for  z,  has  the  form 

z^zi^e-^^'+zi-V-^'+^e-^^'+zfe*^^1'.  (128) 

The  coefficients  of  zj~"  and  zj~°  differ  from  those  of  zf  and  z"'  respectively, 
aside  from  constant  factors,  only  in  the  sign  of  V—  1,  and  these  constant 
factors  differ  only  in  sign.  Hence  it  is  sufficient  to  determine  the  coeffi- 
cients of  z"'  and  z*,  which  have  the  form 


t-o  f-a 

m  I 

y, 


o  ^ 


(129) 


135.  Terms  Independent  of  e. — Equations  (128)  and  (129)  are  to  be 
substituted  in  (127)  and  the  coefficients  of  g**^7'  and  e***^'  set  equal  to 
zero.  These  terms  are  power  series  in  e  whose  coefficients  separately  must 
be  set  equal  to  zero.  We  are  now  interested  in  the  terms  which  are  inde- 
pendent of  e.  These  results  were  worked  out  in  §129,  where  the  parts  of 
x,  and  yt  independent  of  e  were  derived.  The  explicit  results  were  given 
in  equations  (103),  the  relations  in  the  present  notations  being 

.(1)  _<!>     —..(1)  _0>     _/v<» 

it  >  °i.«^  —  Ti  • 


266  PERIODIC   ORBITS. 

The  condition  for  the  consistency  of  the  two  expressions  for  7"'  is  equation 
(106),  which  determines  7"'  =  Co  except  as  to  sign.  Therefore,  by  §132, 
a"}.!  and  $",!  are  also  determined  except  as  to  sign,  and  d,  is  defined. 
And  by  §133  it  is  seen  that  the  C,  ,  $">p  (p^O)  are  all  determined  except 
as  to  sign,  while  the  aj",0  remain  as  yet  undetermined.  The  equations  corre- 
sponding to  (103)  determine  733)  uniquely,  but  7,"  remains  so  far  arbitrary. 

136.  Coefficients  of  e.  —  We  shall  write  explicitly  only  the  terms  which 
involve  those  coefficients  a(^,v,  /8jJi.p,  d,>  and  ^tij>  which,  at  the  successive 


steps,  are  unknown.  The  quantity  zs  is  involved  in  the  right  member 
of  (127)  under  the  integral  sign,  but  since  this  term  is  multiplied  by  e,  it 
introduces  at  this  step  only  C.0  as  an  unknown  coefficient,  and  this  unde- 
termined constant  enters  linearly.  It  is  determined  from  the  terms  which 
are  independent  of  e  when  v  =  2. 

Consider  first  the  parts  of  the  coefficients  of  e*aV=It  and  e4a)N/=IT(  which 
are  independent  of  sint  and  cos<.     We  find  from  (127)  that 


~  2A  a}"  „  a,*},,,  =  2^4  C0a3300+/3'J0,   (130) 


where  f(t~$  and  /£}.„  are  linear  functions  of  C0,  which  is  the  only  un- 
known that  they  involve.  Since  C"  =  —  C,0  ,  the  condition  that  equations 
(130)  shall  be  consistent  is  a  condition  on  their  right  members  which  a 
detailed  discussion  shows  uniquely  determines  the  coefficient  C0-  Then 
equations  (130)  uniquely  define  a®t0. 

Now  we  set  equal  to  zero  the  coefficients  of  e2a)v/:rT'  sint,  e*uV=*'  sint, 
&uV=Itcost,  and  etwv^'cost.  The  explicit  expressions  are  found  from  equa- 
tions (127)  to  be,  respectively, 

VI  [+C.0>  «&  +  4  VA  C»  j 

VA  [  -  C0  &  ~  2  VJ  C0    u  , 


where  ft  ,  .  .  .  ,  <p4  are  functions  of  known  quantities  and  the  arbitrary 
Co>  which  enters  linearly.  This  constant  remains  undetermined  until 
the  equations  are  derived  for  v  —  1.  The  unknowns  in  the  left  members  of 
(131)  are  af^,  f}™.u  d>  and  /3j">u  which  enter  linearly.  On  making  use  of 
the  fact  that  C"  =  —  a"o,0  ,  the  determinant  of  their  coefficients  becomes 

A=-4'(«C)4[4A-1],  (132) 

which  is  distinct  from  zero.  Therefore  these  quantities  are  uniquely  deter- 
mined as  linear  functions  of  the  arbitrary  C,0  • 

This  illustrates  sufficiently  the  method  of  determining  the  coefficients 
from  the  integral.  The  complexity  of  the  details  makes  it  unprofitable  to 
carry  the  explicit  results  further. 


OSCILLATING    SATELLITES,    ELLIPTICAL   CASE.  267 


DIRECT  CONSTRUCTION  OF  THE  TWO-DIMENSIONAL 
SYMMETRICAL  PERIODIC  SOLUTIONS. 

137.  Terms  in  Xv>. — It  was  shown  in  §116  that  the  periodic  solutions 
i  \i-t  and  are  expansible  as  power  series  in  Xv>.  It  is  found  from  equations 
(13)  that  the  terms  of  the  first  degree  in  X1'1  are  defined  by 


x"-2y(-[l+2A+6Aecost+  •  •  •]xl-[6Aesmt+ 
y'+2xfl-[6Aeiant  +  •  •  •}xl-[l-A-3Aecost+ 


(133) 


The  general  solutions  of  equations  (133)  are  known  from  the  general 
theory  to  have  the  form 


(134) 


x,  =  a 
y, =a 
a  =a 

e  +  •  •  •  (t-1 ,4), 


where  a"',  .  .  .  ,  a"'  are  arbitrary  constants,  and  where  the  ut  and  the  vt  are 
periodic  functions  of  t  with  the  period  2r. 

In  order  that  the  solutions  shall  be  periodic  we  must  first  impose  the 
conditions 

aiu=aiu  =  0.  (136) 

The  constant  a  is  a  continuous  function  of  n,  /u,,  and  e.  It  will  be  supposed 
that  these  parameters  have  such  values  that  a  is  a  rational  number.  Then, 
since  u\"  and  t>40)  are  periodic  with  the  period  2*-,  the  solution  at  this  step 
is  periodic  with  the  period  T,  where  T  is  a  multiple  of  2r  and  2*  /<r. 

In  the  symmetrical  solutions,  x'  (0)  =  y  (0)  =  0.  Since  these  relations 
are  identities  in  X1,  we  have  x(  (0)  =  j/,  (0)  =  0.  Since  in  the  symmetrical 
orbits  yi  changes  sign  with  a  change  of  sign  of  t  while  its  numerical  value 
remains  unaltered,  we  have 


(  -  0 
»t(  -  0  -  ai"*-"  v,(  -t)-  a™*»v<(  -  0  . 


It  follows  from  this  identity  that 


268  PERIODIC    ORBITS. 

Without  restricting  the  generality  of  the  results,  we  may  suppose  that 

f i  (0)  =  ^2  (0)  =  v,  (0)  =  v4  (0)  =  1 .     Therefore  we  have 


a"'=  —  a31>(  =  0  in  case  of  periodic  orbits), 


[ept  u,  - 


i=  +<  [e'^'w,-  e-"^'  vt~\+dSi[ef*v1  -e'"1  vj  . 


(136) 


We  shall  suppose  that  the  initial  conditions  are  real  as  well  as  such 
as  to  give  the  symmetrical  orbits.  Then,  since  changing  the  sign  of  V—  1 
in  (133)  does  not  alter  these  equations,  with  the  same  initial  conditions 
the  solutions  will  be  identical  with  (136).  Therefore  we  have 


If  we  change  the  sign  of  both  t  and  yl  ,  equations  (133)  are  unaltered. 
With  the  same  initial  conditions  as  before,  which  this  transformation  does 
not  affect,  since  ?/i(0)  =  0,  we  have  an  identical  solution  except  that  yl  is 
changed  in  sign.  Therefore 


Now  if  V—l,  t,  and  y^  are  changed  in  sign  the  differential  equations 
are  unchanged,  and  hence  it  follows  that 


u4(  —  V—l,  —  f)  =  —u3(V—i,f);         v4(  —  V—i,  —  t)  =  +v3(V—i,f). 

It  follows  from  the  last  three  sets  of  relations  that  u4,  .   .   .  ,  ut 
vlt  .  .  .  ,  vt ,  when  expressed  as  Fourier  series,  have  the  form 


M,  =  S  [ + a,  cos  jt + V  — 1  bj  sinjt] ,  u3  =  S  [  +  c}  cosjt + dj  sin  jt] , 

ut  =  2  [ — a,j  cos  jt + V  —  1  bj  sin  jt] ,  u4  =  S  [  —  c,  cosjt + dj  sin  jt] , 

Vi  =  S  [ + V—  1  ay  cosjt + ft  sin  jt] ,  v3  =  2  [ + y,  cosjt  -f-  5,  sin  $] , 

v i  =  S  [ + V  — 1  a;  cos.# — ft  sin  jt] ,  v4  =  S  [  +  7,  cosjY — 5j  sin  j<] , , 


(137) 


where  the  a/,  fy,  c^,  d;,  a/,  ft,  7,,  and  5,  are  real  constants   and  power 
series  in  e. 


OSCILLATING    SATELLITKS,    KLLIPT1CAL   CASE. 

In  the  case  of  the  |>eriodic  orbits  we  have  simply 


L'li'.t 


(138) 

In  the  ease  of  the  periodic  orbits  it  follows  from  equations  (137)  that  the 
numerical  coefficients  of  the  cosine  terms  in  j,  are  real,  and  that  those  of 
the  sine  terms  are  purely  imaginary;  and  the  opposite  is  true  in  y,. 

138.  Coefficients  of  X.  —  It  is  found  from  equations  (13)  that  these 
coefficients  are  defined  by 


-]:z;J+[-24flesin<-f 


-]zJ+[3fi+12Becos<+ 
+  [9Besmt+ 


.. 
—' 


(140) 


The  character  of  the  solutions  of  the  equations  of  the  type  to  which 
(139)  belongs  was  determined  in  §30,  where  it  was  shown  that  they  consist 
of  the  complementary  function  plus  terms  of  the  same  character  as  Xt  and 
y,  .  Hence  the  periodic  solution  of  (139)  is 


xt  = 
t/,  = 


(141) 


where  ajl>  and  o^*'  are  constants  which  are  as  yet  undetermined,  and  where 
/,  and  0,  are  the  particular  solutions. 

We  shall  need  certain  properties  of  /,  and  gt .  It  is  evident  from  (139) 
and  (140)  that  they  are  homogeneous  of  the  second  degree  in  aj"  and  in 
e*^^7'  and  e~9V=l1.  It  follows  from  (137)  and  (138)  that  in  xj  and  j/J  the 
coefficients  of  those  cosine  terms  which  are  multiplied  by  e**^7'  and  e~"v=:T' 
are  real  and  identical,  while  the  coefficients  of  the  sine  terms  are  purely 
imaginary  and  differ  only  in  sign.  The  terms  which  are  independent  of 
e*^^1'  and  e~rv:rT'  consist  only  of  cosines  whose  coefficients  are  real.  In  the 
product  x,  yl  the  coefficients  of  those  cosine  terms  which  are  multiplied  by 
e"*^'  are  purely  imaginary  and  differ  from  the  coefficients  of  those  cosine 
terms  which  are  multiplied  by  e~tfV={t  only  in  sign,  while  the  coefficients  of 


270 


PERIODIC    ORBITS. 


the  sine  terms  are  real  and  identical.  The  terms  which  are  independent  of 
e<rv=it  an(j  e-ffv/=r<  are  oniy  sme  .terms  and  are  real.  Therefore  it  follows 
from  (140)  that  X2  and  F2  have  the  form 


X,  = 


In  Xf  and  Xj~®  the  coefficients  of  the  cosine  terms  are  real  and  identical, 
and  the  coefficients  of  the  sine  terms  are  purely  imaginary  and  differ  only 
in  sign.  In  X(®  there  are  only  cosine  terms  and  their  coefficients  are  real. 
In  F®  and  F2~2)  the  coefficients  of  the  cosine  terms  are  purely  imaginary 
and  differ  only  in  sign,  and  the  coefficients  of  the  sine  terms  are  real  and 
identical.  In  F20)  there  are  only  sine  terms  and  the  coefficients  are  real. 

It  follows  from  the  properties  of  X2  and  F2  which  have  just  been  derived, 
and  from  the  form  of  equations  (139),  that/2  and  </,  have  the  form 


(142) 


where  /<",  /<~2),  / 


™ 


* 


t/j"2',  and  g(®  are  periodic  with  the  period 


It  follows  from  the  properties  of  Xt  and  F2  which  have  been  found  that 
equations  (139)  are  not  changed  if  in  them  the  sign  of  V—  1  is  changed. 
Therefore  this  property  is  true  of  the  particular  solutions,  and  we  have 


It  also  follows  from  the  properties  of  X2  and  F2  that  if  we  change  the 
sign  of  t  and  y2  ,  equations  (139)  are  not  altered.     Therefore 


It  can  be  proved  similarly,  from  a  consideration  of  equations  (139),  that 


2  [djcosjt  -  V=i  hsinjt],      0J-"  =  2  [-  </= 
2  c,cmjt,  g» 


OSCILLATING    SATELLITES.  ELLIPTICAL   CASE.  271 

It  follows  from  these  three  sets  of  relations  that  when  /"',  .  .  .  ,  g™ 
arc  written  as  Fourier  series  they  have  the  form 


(143) 


where  the  a, ,  b, ,  c, ,  a, ,  0t ,  and  y,  arc  real  constants  and  power  scries  in  e. 

It  is  seen  from  equations  (142)  and  (143)  that  0,(0)  =0.      Therefore, 
since  t/,(0)  =  0,  we  have  a£}  =  — a|l),  and  equations  (141)  become 


<144) 


where  both  a1"  and  oj"  are  so  far  undetermined  constants. 

139.  Coefficients  of  XVl. — It  is  found  from  equations  (12)  and  (13)  that 
these  terms  are  defined  by 


x'-2y'3-  [ l+2A+6Ae  cos*+  •  •  •]  x,-  [6Aesmt+ 

,  /        r  _  .  "I  I"  ,  n   , 

yt  +2z,—  I  vAe  smt+  •  •  •  \  xt—  ^  1~ A  —  3Ae  cost  + 

A  =+[-2K-6Kecost+  •  •  •]*,+  [-6Ke  sinl  + 
+  [  —6B  —  24Becost+  •  •  -]x,x,+  [— 24Besin<+  • 

•     I     i    4*1     •  1      l     i      F  f\  J*i    i  4 

(145) 


[-24flesin<+  •  •  •]xlxt+[ZB+l2Becoat+  •  •  •](xly1+xty1) 
[l8Besml+---]ylyt+[-GC+--]x\yl+[3C+-  •  -]y\; 


v  r  -^    , 

A.  =  -»i  —  J»ri  '  */  =    _«»»    T 

" 


t 


It  follows  from  the  results  of  §29  that  in  general  the  terms  of  the  first 
degree  in  e'^*'  and  e"**^7'  will  introduce  non-periodic  terms  into  the  solution. 
We  must  determine  a{",  if  possible,  so  as  to  make  their  coefficient  vanish. 


272  PERIODIC    ORBITS. 

The  general  solution  of  the  first  two  equations  of  (145),  when  X3  and  Y3 
are  zero,  is 


x3  =  a™ 


"' 


~pt 


(146) 


where  aj3),  .  .  .  ,  a"'  are  arbitrary  constants.  Now,  on  supposing  they  are 
variables  and  subjecting  them  to  the  conditions  that  equations  (142)  shall 
be  satisfied  when  their  right  members  are  included,  we  get 


o^—i 


+ep'[pu3+u'3] 


CO'  =  0, 


(147) 


where  (oJ3))')  •  •  •  >  W)'  are  the  derivatives  of  a{3>,  .  .  .  ,  a"'  with  respect  tot. 
The  solutions  of  these  equations  for  (af)'j  •  •  •  ,  WY  are 


A(a[3')'  =  [Dn  Xt+Da  Y3]  e'"^  ,      A(a«')'  =  [A, 


e'"1  , 


(148) 


where 


A  = 


MI 


o— 


pu3+u'3,      — 

V3  , 

pV3+  V'3  ,         ~ 


—  o-xA^T  Vt+v'z,     pv3+v'3,     — 


,       u3 

,  V3  ,  Vt 

w^,    pu3+u'3,     —pu,i+u( 


<>-'   II. I. \  I  IN',     -\IKI.t.llO.     Kl.l.ll'l  li'AI.    <    \  273 

l>  and  l>  are  obtained  from  I) .,  and  I).  .  respectively,  by  changing  the  sub- 
script '_'  to  1  and  by  changing  the  sign  of  \  ]  and  of  the  whole1  expression: 
/>j,  and  DK  are  obtained  from  l)n  and  ]),,,  resjx>ctively,  by  changing  the 
subscript  .'{  to  1 .  p  to  a  \  -  i .  and  by  changing  the  sinn  of  the  whole  expre— 
sion:  and  />„  and  />.,are  obtained  from  I),  and  I),.,  respectively,  by  changing 
the  subscript  I  to  1,  —  ptoff\  I .  and  by  changing  the  sign  of  the  whole 
expression.  It  follows  from  the  di.-cu-Mon  of  §  is  that  A  is  a  constant,  and 
in  this  case  it  is  a  power  series  in  • . 

In  order  that  the  solution  -hall  be  periodic  it  is  necessary  that  the 
right  members  of  (148)  shall  contain  no  constant  terms.  \Ye  shall  show  these 
conditions  are  sufficient.  \Yhen  they  are  satisfied  the  general  term  of  the 
riuht  member  of  either  of  the  first  two  equations  has  the  form 

[a,.»  cosjt+bjj  slnjt]  e**"^^', 

where  j  and  /.  are  integers  distinct  from  xero  and  where  n,k  and  bJL  are  con- 
stant-. <  '.-iiseijuently  a?  and  a*  are  sums  of  terms  of  the  t  \  pe 


•V^itAnjt]  ^ijacf  (149) 


The  right  member  of  the  third  equation  of  (148)  never  has  any  terms 
which  are  independent  of  I,  but  contains  terms  of  the  type 


.•os/7  rAtriiyf 


[(»fU<rV^T-< 


where./  and  *  are  integers.     There  can  be  no  exception  to  tins  form.     There- 
fore "     i-  a  sum  of  terms  of  the  type 


(150) 


The  type  terms  for  a?  differ  from  those  for  a?  only  in  the  sign  of  p.  There 
is  an  additive  constant  of  integration  with  each  of  the  a',3'.  It  follows,  from 
the  form  of  (146),  (149).  and  (150),  that  if  we  put  the  constants  of  integra- 
tion associated  with  //;  and  n"'  equal  to  zero,  the  resulting  expressions  for 
xa  and  y,  are  i>eri<)dic  with  the  jjeriod  T.  They  may  be  written  in  the  form 


\\here  aj"  and  aj11  arc  constants  which  so  far  are  undetermined. 


274 


PERIODIC    ORBITS. 


It  remains  to  show  that  a"'  can  be  so  determined  that  the  right  members 
of  the  first  two  equations  of  (148)  shall  have  no  constant  terms.  Let  us 
consider  the  first  of  these  equations.  We  are  to  set  equal  to  zero  the  con- 
stant part  of  the  coefficient  of  e"v^'  in  DnX3+DaY3.  It  follows,  from  the 
form  of  X3  and  Y3,  equations  (145),  that  the  term  which  must  be  made  to 
vanish  does  not  depend  on  af\  It  also  follows  that  the  conditional  equa- 
tion which  must  be  imposed  has  the  form 

where  Pl  and  Qi  are  power  series  in  e,  the  former  coming  from  those  terms 
of  X3  and  F3  which  are  linear  in  x^  and  y^  and  independent  of  x2  and  yt, 
and  the  latter  coming  from  those  terms  which  are  of  the  third  degree  in 
xl  and  yl ,  or  which  involve  x2  and  y2. 

The  solutions  of  (152)  are  a{"  =  0,  which  leads  us  to  the  trivial  result 
x=y=Q,  and 

In 

(153) 


The  significance  of  the  double  sign  was  discussed  in  §§116-118  in  connection 
with  the  existence  of  the  solutions.  The  expressions  for  Pt  and  Qi  are 
power  series  in  e  and  both  of  them  contain  terms  independent  of  e,  as  was 
shown  in  Chapter  VI  in  the  discussion  of  the  corresponding  problem  for 
e  =  0.  Therefore  crjl)  is  a  power  series  in  e  having  an  absolute  term. 

It  remains  to  be  shown  that  this  value  of  a™  also  satisfies  the  equation 
which  is  obtained  when  the  constant  term  of  the  right  member  of  the  second 
equation  of  (148)  is  set  equal  to  zero.  We  shall  show  that  the  constant 
part  of  the  coefficient  of  eav^'  in  Dn  X,  +  D12  Y3  is  identical  with  the  con- 
stant part  of  the  coefficient  of  e~av=~-1  in  Z)21X3+DM  Y3 .  Let  us  first  consider 
the  term  [— 2K— QKecost-}-  •  •  •  ]xl  of  Xt  which  contributes  to  Pl  of  equa- 
tion (152).  So  far  as  this  term  is  concerned,  we  have 


(154) 


On  referring  to  (138)  and  the  values  of  Dn  and  D21,  we  see  that  the  constant 
parts  of  the  right  members  of  these  two  equations  are  respectively  the 
constant  parts  of 


-a 


u. 


-< 


,     v,  »« 

,    pVt  +  Vf,,      -pt>4  +  t>4 


-pvt+v'4 


[-2K-6Kecost-\ 


[-2K-6Kecost-\ 


(i>(  II. I. \  IIM.    >\IKI.I.m.-      M.I.II'IK   \L   CASE. 

These  expressions  arc  equivalent  to 


275 


MI' 


t>,-t>4 


whore 


The  parts  of  these  expressions  containing  M,M,  as  a  factor  are  identical 
ami  need  no  further  consideration.  The  parts  multiplied  by  »/,«-,  and  u,wlf  so 
far  as  they  apjwar  in  the  second  lines  of  the  determinants,  arc  respectively 


[-2A'-6A'ccos«H J, 


_(!> 

~2~ 


?t,-U, 


[-2/C-6A'ecos<H 


It  follows  from  (137)  that  M,  —  w,  is  a  sum  of  cosine  terms,  that  u,  +  M4  is  a 
sum  of  sine  terms,  that  r,+«',  is  a  sum  of  cosine  terms,  that  v't  —  v't  is  a  sum 
of  cosine  terms,  that  «',— »4  is  a  sum  of  sine  terms,  and  that  v't+t\  is  a  sum 
of  sine  terms.  Therefore  the  determinant  parts  of  these  two  expressions 
are  sums  of  sine  terms,  which,  multiplied  by  a  cosine  series  on  the  right, 
are  sums  of  sine  terms.  Hence,  to  get  the  constant  parts  of  these  expres- 
sions, we  need  only  the  sine  terms  of  the  products  u,t',  and  u,t',.  It  is  seen 
at  once  from  (137)  that  the  sine  terms  of  these  products  are  identical,  but 
that  the  cosine  terms  differ  in  sign.  Therefore  the  constant  terms  coming 
from  the  parts  of  the  two  expressions  which  are  multiplied  by  M,r,  and  n,t>,, 
so  far  as  they  come  from  the  second  lines  of  the  determinants,  are  identical. 
The  parts  of  the  expressions  which  contain  u,t>,  and  «,t>, ,  so  far  as  they 
come  from  the  third  lines  of  the  determinants,  are  respectively 


tt,-U4 


V,  ~ 


The  determinant  is  in  this  case  a  sum  of  cosine  terms.    Therefore  we  need 
only  the  cosine  terms  from  -fttif,  and  —  u^r, .     It  is  seen  from  (137)  that 


276 


PERIODIC    ORBITS. 


they  are  identical      Therefore  the  constant  parts  of  the  two  expressions,  so 
far  as  they  arise  in  this  manner,  are  identical. 

It  remains  to  consider  only  the  constant  parts  of  the  two  functions 


-  *>4 


U3-U4, 


[-2K-6Kecost+ 


[-2K-6Kecost+ 


It  follows,  as  before,  that  we  need  only  the  cosine  terms  of  —  wX  an(i  — 
We  see  from  (137)  that  the  coefficients  of  the  cosine  terms  in  these  products 
are  identical.  Therefore  the  constant  parts  of  the  right  members  of  equa- 
tions (154)  are  identical. 

The  discussion  for  the  other  terms  of  X3  and  Y3  which  are  linear  in  xl 
and  ?/!  is  made  in  a  similar  manner,  and  it  is  thus  proved  that  the  Pl  which  is 
obtained  from  the  second  equation  of  (148)  is  identical  with  the  one  which 
depends  on  the  first. 

It  is  now  necessary  to  consider  those  terms  of  X3  and  Y3  which  are  not 
linear  in  xl  and  yl .  Let  us  treat  in  detail  the  term  in  X3  which  contains 
xlxt  as  a  factor.  So  far  as  this  term  is  concerned,  the  first  two  equations 
of  (148)  become 


] 


-oV—  if 


(155) 


On  referring  to  equations  (138),  (144),  and  the  expressions  forZ)n  andD21,  it 
is  seen  that  the  constant  parts  of  the  right  members  of  these  equations  are 
respectively  identical  with  the  constant  parts  of 


U3 


v't,  pV3+V3,   - 


*>4 


-K')3 


M! 


1*4 
»4 


pv3+v'a,  - 


(-GB-24Becost+ 


Since  f(®  is  a  cosine  series,  and  the  product  of  it  and  [  —  65  —  24  Be  cos  t-\-  •  •  •] 
is  also  a  cosine  series,  the  discussion  for  the  terms  multiplied  by  Wj  /20)  and 
Ut  /i0)  does  not  differ  from  that  given  above  for  the  terms  multiplied  by  x^ . 


OSCILLATING    SATELLI'l  1>.   KLLIPTICAL   CASE. 

\\ 'c  have  now  to  find  the  constant  parts  of 


K')' 
a 


K')' 
2 


277 


-v't,  p(vt-vt)+v't+v't 


i.-,<; 


where  /•',(*)  =  [-6B-24Becost+  •  •  •  ]. 

The  factors  by  which  ti^/f  and  MJ/^C  are  multiplied  in  these  respec- 
tive expressions  are  identical,  and  it  follows  from  equation  (137)  that  they 
are  a  sum  of  cosine  terms  having  real  coefficients.  Consequently  we  need 
only  the  cosine  terms  of  wj  f*  and  wj/i"1'  in  order  to  obtain  the  constant 
parts  of  (156).  Now  it  follows  from  (137)  and  (143)  that  the  cosine  terms 
of  the  products  wj/i"  and  w?/i~B  are  identical.  Therefore  the  constant 
parts  of  (156),  which  involve  H|  and  u\  as  factors,  are  identical. 

Now  consider  r,  and  vr  in  so  far  as  they  occur  in  the  second  lines  of  the 
determinants.  It  follows  from  (137)  that  the  factors  by  which  —  w,r,  /" 
and  —  J^fi/i"*'  must  be  multiplied  are  sine  series  having  real  coefficients. 
Therefore  we  need  only  the  sine  terms  in  these  products.  It  also  follows 
from  (137)  that  the  expressions  for  n^vt  and  w,v,  are  respectively  cosine 
terms  with  purely  imaginary  coefficients  which  differ  only  in  sign,  and  sine 
terms  with  real  coefficients  which  are  identical.  Therefore,  in  the  products 
MjVj/J"  and  u^vj^  the  coefficients  of  the  sine  terms  are  real  and  respec- 
tively equal. 

There  remain  only  the  terms  coming  from  the  third  line  and  first  column 
of  the  determinants.  We  have  first  —  aV  —  1  w,t;t/,U)  and  -f  vV—  1  utvj^. 
These  expressions  are  multiplied  into  the  same  cosine  series  having  real 
coefficients.  Consequently  we  need  compare  only  the  coefficients  of  their 
n»ine  terms,  which  we  find  from  (137)  and  (143)  are  real  and  respectively 
identical.  Therefore  the  constant  parts  of  the  right  members  of  (155)  to 
which  these  terms  give  rise  are  identical. 

Finally,  there  remain  only  the  terms  multiplied  by  +M,t>i/Jn  and  by 
+wX/i~l)  respectively.  The  term  into  which  these  factors  are  multiplied 
is  a  cosine  series  having  real  coefficients.  It  is  seen  from  (137)  and  (143) 
that  the  coefficients  of  the  cosine  terms  of  +t/X/,a>  and  -fwX/«~c  are  real 
and  respectively  identical.  Therefore  the  constant  parts  of  the  right 
members  of  (155)  are  altogether  identical. 

The  discussions  for  all  the  other  terms  of  A',  and  1',  which  involve  x, 
or  t/,  are  made  in  a  similar  manner  and  lead  to  the  same  result.  There 
remain  only  terms  in  Xt  and  Yt  which  are  of  the  third  degree  in  a;,  and  yt . 


278 


PERIODIC    ORBITS. 


Let  us  consider,  for  example,  the  term  of  A'3  which  is  multiplied  by  x^y}. 
Then,  so  far  as  this  term  alone  is  concerned,  the  first  two  equations  of 

(148)  become 

3''  =         -  .  .  .]x,rf«r'^',l 

= 


The  constant  parts  of  the  right  members  of  these  equations  are  respectively 
the  constant  parts  of 


U3 


,     v, 


pv3+v'3>  - 


ut 


pv3+v'3,  —pvt+v't 


[-6CH 


(158) 


Since  v^vt  is  a  cosine  series  having  real  coefficients,  the  discussion  for  the 
the  terms  multiplied  by  2M,?^2  and  2u2v1vt  does  not  differ  from  that  given 
above  for  that  part  of  X3  which  is  multiplied  simply  by  #, . 

If  we  refer  to  equations  (137)  and  (138),  we  see  that  v\  and  v\  have  the 
properties  of  ff  and  /2~2>,  as  regards  the  relations  existing  between  their 
respective  coefficients.  Therefore  the  discussion  of  these  terms  of  (158)  is 
identical  with  that  of  (156),  for  which  the  proposition  was  established. 

In  a  manner  similar  to  this  the  identity  of  the  constant  parts  of  the 
right  members  of  the  first  two  equations  of  (148)  can  be  established  for  all 
of  the  elements  of  which  X3  and  F3  are  composed. 

140.  General  Proof  that  the  Constant  Parts  of  the  Right  Members  of 
the  First  two  Equations  of  (148)  are  Identical. — We  shall  treat  first  the 
parts  which  depend  on  X3 .  We  shall  need  the  following  properties  of  X3 : 

(1)  It  is  a  polynomial  in  x1}  ylt  z2,  y2. 

(2)  Those  terms  which  are  of  even  degree  in  yl  and  ?/2  taken  together 
are  multiplied  by  cosine  series  having  real  coefficients. 

(3)  Those  terms  which  are  of  odd  degree  in  yl  and  t/2  taken  together  are 
multiplied  by  sine  series  having  real  coefficients. 

(4)  If  the  general  term  is  XiX^y^y^-,  then  ji+2j2-}-kl-}-2k,  is  an  odd 
integer  (in  the  present  case  one  or  three). 

The  parts  of  the  first  two  equations  of  (148)  which  depend  on  X3a,re 

It  is  obvious  from  (137)  and  properties  (2)  and  (3)  that  those  parts  of 
X3  e~av/~'  and  X3  effv~'  which  are  independent  of  the  exponentials  e~av^' 
and  e**'^7'  are  sums  of  cosines  having  real  coefficients  and  of  sines  having 
purely  imaginary  coefficients,  and  that  the  real  coefficients  in  the  two  ex- 


OSCILLATING    SATELLITES,    ELLIPTICAL   CASE. 


279 


pn -sinus  differ  respectively  only  in  sign,  while  the  imaginary  coefficients  are 
respectively  identical.  Hence,  referring  to  the  expressions  for /.),,  :iml  I)  . 
we  may  write  these  parts  of  equations  (159)  in  the  form 


r, 


U, 


t>,-t>4 

'»,  P(v*+v4)+v',-v'< 


, 


2[A,cosjt 


U,-U4 


v{,  P(vt+v4)+vt-v'4l 


It  easily  follows  from  these  expressions,  as  in  the  discussion  in  §  139,  that  their 
constant  parts  are  real  and  identical. 

Now  consider  the  terms  depending  on  Ytl  which  has  the  properties 
(1)  and  (4)  belonging  to  A", ,  and 

(2)  Those  terms  which  are  of  even  degree  in  yl  and  yt  taken  together  are 
multiplied  by  a  sine  series  having  real  coefficients. 

(3)  Those  terms  which  are  of  odd  degree  in  yl  and  yt  taken  together  are 
multiplied  by  a  cosine  series  having  real  coefficients. 

The  parts  of  the  first  two  equations  of  (148)  which  depend  on  F,  are 

It  follows  from  (137)  and  properties  (2)  and  (3)  that  those  parts  of  F,  g-»v=ri 
and  F,  e*yrr7'  which  are  independent  of  the  exponentials  e~*vrrT'  and  e9^^' 
are  sums  of  cosine  terms  having  purely  imaginary  coefficients,  and  of  sine 
terms  having  real  coefficients,  and  that  the  purely  imaginary  coefficients  are 
respectively  identical  while  the  real  coefficients  differ  respectively  only  in 
sign.  Hence,  using  the  explicit  values  of  Da  and  Da,  these  parts  of  the 
right  members  of  (160)  are  found  to  have  the  form 


w,-u4 
vt-v4 


^lAi  cosjt 
+B,smjt], 


—  B,8injt]. 


It  follows  from  (137)  that  the  constant  parts  of  these  two  expressions  are 
real  and  identical.  Therefore  the  constant  parts  of  the  right  members  of  the 
firsl  two  equations  of  (148)  are  identical,  and  when  one  of  them  is  made  to  vanish 
by  a  special  determination  o/a|"  the  other  one  also  vanishes. 


280 


PERIODIC    ORBITS. 


141.  Form  of  the  Periodic  Solution  of  the  Coefficients  of  XVl. — It  follows 
from  the  form  of  X3  and  Y3,  given  in  equations  (145),  that  /,  and  g3  of  equa- 
tion (151)  have  the  form 


g? 


g 


,,(0) 


(-3)      -3 

5  t/ 


(161) 


J 


where  /"'  ,  .  .  .  ,  <?s  3)  are  known  functions  of  t .  We  need  certain  properties 
of  these  functions.  It  follows  from  the  properties  of  X3  and  F3  and  of  the 
left  members  of  the  differential  equations,  and  from  certain  considerations 
of  changes  of  sign  of  V^l,  t,  and  y3,  in  both  the  differential  equations  and 
the  solutions,  that 


g(3n(  —  V^T,  —  t)  =  —  g^    (V^T,  t)  O'  =  0,  1,  2,  3,  -1,  -2,  -3). 

It  follows  from  these  relations  that  the  /"'  and  the  g'f  have  the  form 


/l-3)  =  2  [af  cosjt- 
/f   =  2  [<  c 
/<-2)  =  2  [af  cosji- 


=7&«sin#],         0i-B=Z[ 
sinjZ],         ^   =2 
^T&fsiiyY],         <73~2>  =  2 


cosjt+0™  sinjt], 


cosjt+ff  smjt], 


sin^], 


/3-1)=2[-v^T<)cosj7+^ 


™ 


-,(0) 


(162) 


/i-B-2(ofcosj<-V=i 

/<»   =2  o«'cos^, 

where  the  af ,  .  .  .  ,  /3j"  are  real  constants. 

It  follows  from  equations  (161)  and  (162)  that  </3(0)=0.     Therefore, 
since  y3(0)  =  0,  we  have  «"'  =  —  af ,  and  equations  (151)  become 


(163) 


142.  Coefficients  of  X2.—  -It  is  seen  from  (]'.{)  (hat  the  coefficients  of 
X2  are  defined  by  the  differential  equations 


/,-  A', 


.<-   II.I.VHV.    >\IKU.m>.    Kl.l.IITK-AI,    f\ 

where 

\'4=+[-2K-(\Kcco8t+  •  •  •]*,+[  -(iAYsin<  +   • 


I'M 


[-24fiesin<+  •  •  •  ]  [^ 
[--j/H-ttlfeco8H-  •  • 


[ZB+2Beco*t+ 


(165) 


\\  It-  -re  X4  and  Y4  are  independent  of  x,  and  j/,  and  linear  in  x,  and  j/,  .  In 
.V,  tin  in  ins  which  are  of  even  degree  in  yl  and  j/t  are  multiplied  by  cosine 
M-rirs  having  real  coefficients,  while  those  which  are  of  odd  degree  in  yl  and 
y,  are  multiplied  by  sine  series  having  real  coefficients.  In  the  case  of  Yt 
tin-  cosine  series  and  sine  series  are  interchanged.  If  z{'  xj  xj  yj'  j/J'  yj1 
is  tin-  -i  ncral  term  in  X4  or  1'4  ,  then  jt+2jt+3j,  +  kl  +  2kt+3kt  =  4  or  2. 
\Yhon  the  right  members  of  (164)  are  set  equal  to  zero,  the  general  solution 
of  the  equations  is 


where  a\ 


*> 


aj4*  are  arbitrary  constants.     Now,  on  varying  them  and 


subjecting  them  to  the  conditions  that  (164),  including  the  right  members, 
>li:ill  bo  satisfied,  we  find 


A  «)f-[fl,,  A'4+  Da  1'J 


A 

A 


whore  D 


;,  are  the  same  as  in  §139. 


\rrr--ury  conditions  that  the  solution  shall  be  periodic  at  this  stoj)  are 
that  the  constant  terms  in  the  right  members  of  the  first  two  equations  of 
!  KIT  '  >hull  IK>  zero.  It  follows  from  the  form  of  A'4  and  Y,  ,  as  given  in  (165), 
:iiid  from  their  properties,  that  these  constant  terms  are  independent  of 

.nd  involve  n\-'  linearly.  Therefore  the  condition  that  the  constant 
trim  of  the  right  member  of  the  first  equation  shall  be  zero  has  the  form 


(168) 


282 


PERIODIC    ORBITS. 


where  P2  and  Q2  are  power  series  in  e.  It  was  shown  in  Chapter  VI,  in  the 
treatment  of  the  case  where  e  =  0,  that  P2  has  a  term  independent  of  e  which 
is  distinct  from  zero.  Therefore  for  \e\  sufficiently  small  a®  is  uniquely 
determined  by  (168)  as  a  power  series  in  e. 

The  equation  obtained  by  setting  the  constant  part  of  the  right  member 
of  the  second  equation  of  (167)  equal  to  zero  is  of  the  same  form  as  (168); 
it  is,  in  fact,  identical  with  (168),  as  will  now  be  shown.  It  follows  from 
the  properties  of  Xt  and  Y4  that  the  parts  of  the  right  members  of  the 
first  two  equations  of  (167)  which  are  independent  of  the  exponentials 
e-<rv=T<  an(j  girv^Ti  have  the  form 


cosjt+D}  sinjt], 
^T  C,  cosjt+Dj  sinjt]. 


(169) 


These  equations  are  of  exactly  the  same  form  as  those  encountered  in  §  140 
in  the  preceding  step  of  the  integration,  and  the  conclusion  follows  in  the 
same  manner.  Consequently  if  a®  is  determined  so  as  to  satisfy  (168),  and 
if  the  additive  constants  arising  with  the  integrals  of  the  last  two  equations 
of  (167)  are  taken  equal  to  zero,  then  the  solutions  of  (164)  are  periodic.  It 
follows  from  the  properties  of  Xt  and  F4  that  they  have  the  form 


z4  =  a?VA 


(-2>e-2ffv/=l 


+a*f*  + 


+4 


-4 


(170) 


where  the  /"'  ,  g\"  ,  /"}  ,  and  g[1}  have  properties  exactly  analogous  to  those 
of  equations  (162). 


143.  Induction  to  the  General  Term  of  the  Solution.  —  We  shall  suppose 


the  xl 


£„_,;  ?/, 


yn-i  have  been  computed  and  that  their 


coefficients  have  all  been  determined  except  a,""2'  and  aj"~",  which  enter  in 
xn-2  ,  yn-2  ,  xn-i  ,  and  i/n_,  in  the  form 


=*  v.-e^^^'  vt] 


(171) 


OSCILLATING   SATELLITES,   ELLIPTICAL  C.\  283 

\\V  shall  suppose  that  x,  and  y,  (p=  1,  .  .  .  ,  n-1)  have  the  properties 
x,    =   2  /y'e"^7',  y,  =    2'  0i" 


(172) 


2[+v'=7a,*-')co8»-<+#r-"8in>/i] 


The  differential  equations  which  define  xn  and  y.  are  seen  from  (13)  and 
(14)  to  be 


•]z.-[6v4esin<+  •  •  -]y.  =  X., 


y'+2xn-[6Ae  smt+ 
where 


(173) 


+[-3B-12Becos(+ 


xn_t+(K+3Kecost+ 


(174) 


-   '  '  •][2yly._1+2y1y._,]+r4. 

The  functions  X,  and  Yn  do  not  involve  z._i  or  y._i ,  and  are  linear  in  z._t 
and  y._,.  In  X.  the  terms  which  are  of  even  degree  in  y,  ,  .  .  .  ,  y,_, 
are  multiplied  by  cosine  series  having  real  coefficients,  while  those  which  are 
of  odd  degree  in  y, ,  .  .  .  ,  y._»  are  multiplied  by  sine  series  having  real 
coefficients.  In  the  case  of  Ym  the  cosine  series  and  sine  series  are  inter- 
changed. Ifz{'  •  •  •  zi'-V  yf1  •  •  •  yi"_V  is  the  general  term  of  X,  or  Yu,  then 

^i+2j,+  •  •  •  +(n— l)j,_,-}-A^4-2^+  •  •  :  +(n—  !)&„_,  =  n  or  n-2.     (175) 

Necessary  conditions  that  the  solutions  of  (173)  shall  be  periodic  are 
that  the  right  members  of 


shall  contain  no  constant  terms.  It  follows  from  (174)  that  these  constant 
terms  are  independent  of  a"""  and  involve  aj"~"  linearly.  The  coefficient 
of  a|"~"  is  distinct  from  zero  for  |e|  sufficiently  small,  for  in  Chapter  VI  it 


284 


PERIODIC    ORBITS. 


was  seen  to  be  distinct  from  zero  for  e  equal  to  zero.  Therefore  af"2'  is 
uniquely  determined  as  a  power  series  in  e  by  setting  the  constant  term  of 
the  right  member  of  the  first  equation  of  (176)  equal  to  zero. 

It  can  be  shown,  precisely  as  in  the  discussion  when  w  =  4,  that  the 
constant  parts  of  the  right  members  of  equations  (176)  are  identical.  There- 
fore af~2>  is  uniquely  determined  by  the  conditions  that  the  solutions 
of  (173)  shall  be  periodic.  It  follows  from  the  properties  of  xi}  .  .  .  ,  xn^; 
y1 ,  .  .  .  ,  yn-i,  and  from  (175),  that  when  these  conditions  are  satisfied  the 
solution  of  (173)  has  the  form 


=  /,<•)  0aV= 


{"-1)  [g 


o-Pt  u 

+» 

S 

j=-n 

—  +» 


(177) 


(n) 


aj"  °  are  undetermined,  and  where  the  /"',  /),~2>,  /10), 


where  a[B),  .  .  . 

^.a),  On"",  9?,f",  and  ^'  have  the  properties  of  (172). 

In  order  that  (177)  shall  be  periodic  it  is  necessary  and  sufficient  to 
impose  the  conditions  a™  =  a%t>  =  0.     Then  it  follows,  from  the  properties 


of  »!, 
that  o 


,, 


y 


that  y.  (0)  -  of'  -  <  =  0.     Since  y.  (0)  =  0,  it  follows 


=  —  d?\  and  equations  (177)  become 


xn  = 


(-2)  g-2 


+» 
+   S    f" 

I        "     J  n 


+   2 


(178) 


and  equations  (172)  are  satisfied  for  p  =  n. 

In  picking  out  the  constant  part  of  the  right  member  of  the  first  equation 
of  (176),  in  general  only  those  terms  in  Xn  and  Yn  which  contain  ea^/=lt 
as  a  factor  to  the  first  degree  will  be  used.  But  because  a  is  a  rational 
number  there  will  eventually  occur,  in  the  higher  powers  of  the  exponentials, 
multiples  of  a  which  are  integers,  and  constant  terms  in  the  right  member 
of  the  first  of  (176)  may  occur  from  these  terms,  but  their  presence  does 
not  prevent  the  determination  of  the  constants  so  that  the  solutions  shall  be 
periodic.  After  such  terms  once  appear,  they  occur  in  general  at  each 
succeeding  step  of  the  integration. 


CHAPTER  VIII. 

THE  STRAIGHT- LINE  SOLUTIONS  OF  THE  PROBLEM 

OF  H  BODIES. 

144.  Statement  of  Problem.  In  his  prize  memoir*  on  the  problem  of 
time  bodies,  Lagrange  showed  that,  for  any  three  finite  masses  mutually 
attracting  one  another  according  to  the  Newtonian  law,  there  are  four 
distinct  configurations  such  that,  under  proper  initial  projections,  the  radios 
of  the  mutual  distances  remain  constant.  The  bodies  describe  similar  conic 
sections  with  respect  to  the  center  of  mass  of  the  system,  the  simplest  case 
being  that  in  which  the  orbits  are  circular.  In  three  of  the  four  solutions 
the  masses  lie  always  in  a  straight  line,  and  in  the  fourth  they  remain  at  the 
vertices  of  an  equilateral  triangle.  This  memoir  is  one  of  the  most  elegant 
written  by  Lagrange,  and  its  mathematical  form  does  not  seem  capable  of 
improvement.  But  the  method  which  he  employed  can  not  lie  extended 
conveniently  to  the  case  of  more  than  three  bodies,  and  it  has  not  led  to 
practical  results  in  applied  celestial  mechanics. 

This  chapter  is  devoted  to  the  solution  of  two  closely  related  problems: 

I.  The  number  of  straight-line  solutions  is  found  for  n  arbitrary  positive 
masses;  that  is,  the  ratios  of  the  distances  are  determined  so  that  under 
proper  initial  projections  the  bodies  will  always  remain  collinear.     This  is 
the  generalization  of  Lagrange's  straight-line  solutions  to  the  problem  of  n 
bodies.     For  each  straight-line  solution  of  n  finite  masses  there  aro  n-fl 
points  of  libration  near  which  there  are  oscillating  satellite  orbits  of  the 
types  treated  in  Chapters  V — VII.     Therefore  the  results  of  this  chapter 
lead  to  generalizations  of  those  of  the  preceding  three  chapters. 

II.  The  problem  is  solved  of  determining,  when  possible,  n  masses  such 
that,  if  they  are  placed  at  n  arbitrary  collinear  points,  they  will,  under  proper 
initial  projection,  always  remain  in  a  straight  line. 

The  first  problem,  in  a  somewhat  different  form,  has  been  considered 
b\  I>ehmann-Filhes.t  The  method  of  treatment  adopted  heref,  though 
originally  developed  independently,  has  considerable  in  common  with  that 
of  Lehmann-Filhes,  and  the  discussion  completes  in  certain  essential  respects 
the  demonstration  of  the  German  astronomer.  It  is  shown  that  whatever 
real  positive  values  the  masses  may  have,  and  whatever  the  rate  of  their 
revolution,  there  are  $n!  real  collinear  solutions. 

•Lagrange's  Collected  Work*,  vol.  VI,  pp.  229-324.    TuMrand'i  Mttaniqve  CtietU,  vol.  I,  chap.  8. 
\Atlronomitche  Nachrichten,  vol.  CXXVII  (1891).  No.  3033. 

JRead  before  the  Chicago  Section  of  the  American  .Mathematical  Society,  December  28,  1900;  abstract 
in  Bull.  Am.  Math.  Nor.,  vol.  VII  (1900-1901),  p.  249.  tu 


l»N(i  I'KIIIODIC 

Ill  MIC  second  problem  it  is  proved  that  when  the  number  of  arbitrarily 
chosen  real  collinear  points  is  even,  the  n  masses  are,  in  general,  uniquely 
determined  by  the  condition  thai  it,  shall  be  possible  to  place  them  at  these 
points  and  to  give  them  initial  projections  so  that  they  will  always  remain 
collinear  and  revolve  in  orbits  of  specified  linear  dimensions.  Or,  if  it  is 
preferred,  the  period  of  revolution  can  be  taken  as  the  arbitrary  in  place 
of  the  linear  dimensions  of  the  orbit.  In  general,  the  masses  will  not  all  be 
positive,  and  therefore  the  problem  will  not,  always  have  a  physical  inter- 
pretation. When  I  he  number  of  points  is  odd,  it  is  not  possible  to  determine 
the  masses  so  as  to  satisfy  the  solution  conditions  unless  the  coordinates  of 
the  points  themselves  satisfy  one  algebraic  equation.  \Yhen  they  satisfy 
this  condition,  any  one  of  the  masses  may  be  chosen  arbitrarily,  after  which 
all  of  the  others  are,  in  general,  uniquely  determined. 


I.    DETERMINATION  OF  THE  POSITIONS  WHEN  THE 
MASSES  ARE  GIVEN. 

145.  The  Equations  Defining  the  Solutions.  Let  the  origin  of  coordi- 
nates be  taken  a!  the  center  of  gravity  of  the  system,  which  will  be  supposed 
to  bo  at  rest.  This  point  and  the  line  of  initial  projection  of  any  mass 
determine  a  plane.  All  the  other  masses  must  be  projected  in  this  plane, 
for  otherwise  they  would  not  be  collinear  at  the  end  of  the  lirst  element  of 
time.  All  the  bodies  being  initially  in  a  line  and  projected  in  the  same 
plane,  I  hey  will  always  remain  in  this  plane.  Consequently,  if  solutions 
exist  in  which  the  n  masses  arc  always  in  a  straight  line,  the  orbits  arc 
plane  curves. 

Let  the  plane  of  the  motion  be  the  £7;  plane.  Let  the  masses  be  de- 
noted by  mn  m,,  .  .  .  ,  m»,  and  their  respective  coordinates  by  (£,,  TJ,), 
(&>  *?•)»  •  •  •  »  (£»>  '?»)•  Then,  choosing  the  unite  so  that  the  Gaussian 
constant  is  unity,  the  differential  equations  of  motion  are 


<f£,       1    dU  ^u,       1    dU  . 

m,  d^ 


u  -   S  S 


m'  - 


(1) 


Equations  (1)  always  admit  the  integral  of  areas 

"*  H 

(t  ui)t          "<i\        V^  .,    „«  dOt  /TTT — r  f<)\ 

v  ~jf  ~~  >?•  "jj  J          //  '"*i  ^«  "jT  ™  c>  r«      V d  T  >?(  • 


KM'.III    LINK    x.1.1    IK. N>    H)l<    «    BODIKB.  287 

Iii  ca-e  tin-  n  bodie-  remain  collinear,  the  line  of  the  n-ultaiit 
acceleration  to  which  each  one  is  -ubject  alwav-  pa--e-  through  the  origin. 
'I 'herefore.  in  collinear  solution-  it  follows  from  the  law  of  ureas  that,  for 
each  body  separately. 


Hut  when  the  bodie-  remain  collinear  we  have  also 

</«i      dOj  dO. 

dt'    ilt  '  dt  ' 

from  which  it  follows  that 

d~ 

'a./,  (3) 


when-  the  n,,  are  constants.  That  is,  if  any  collinear  solutions  exist,  the 
ratios  of  the  distances  of  the  bodies  from  the  origin  are  constants,  ami  it 
easily  follow-  from  thi-  that  the  ratios  of  their  mutual  distances  are  also 
con -'.in'-  They  are  therefore  of  the  I..agrangian  type. 

If  the  it  masses  remain  collinear,  the  ratios  of  their  distances  from  the 
origin  I.einn  therefore  constants,  the  ratios  of  their  coordinates  are  constants. 
Therefore  in  all  collinear  solutions 

tt-xtt,         n«-*i»?  (»'-! ").         (4) 

where  the  z,  are  constants.  Upon  substituting  in  equations  (1),  we  have, 
as  necessary  conditions  for  the  existence  of  the  collinear  solutions, 


(5) 


In  order  that  £  and  rj  as  defined  by  their  initial  values  and  equations 

hall  }*•  the  same  for  all  values  of  i,  the  coefficients  of  {/r1  and  jj/r1  must 

be  set  Mjual  to  a  constant  independent  of  i.     Letting  —  w*  represent  this 

constant  and  r,,-=  V(xt— x,Y,  these  conditions,  which  are  sufficient  as  well 

as  necessary  for  the  existence  of  the  collinear  solutions,  become 


d%  y    mfa-x,)    i 

df  2*S*-*?rr 

rfS;  V*    m,(xt-X,) 
df 


0 

1l  ^U  ' I  • 

•  *O   .  .   m.fx. —  x.)   .  .   m.(Xt  —  X.)         . 

(6) 


t  mt(xm-Xt)  j  mt(z.-Xt)  |  m>(z.-z>)  | 
>~\  i\*  *•* 


288  PERIODIC    ORBITS. 

Suppose  the  notation  is  so  chosen  that  in  any  solution  xl  <  •  •  •  <  xn  ; 
then  the  terms  of  the  left  member  of  the  last  equation  are  all  positive. 
Since  the  origin  is  at  the  center  of  gravity,  xn  is  positive,  and  therefore  a"  is 
positive  in  all  real  solutions.  For  every  set  of  values  of  x1  ,  .  .  .  ,  xn 
satisfying  equations  (6)  the  solutions  of  (5)  are  the  same  for  all  values  of  i, 
and  these  solutions  substituted  in  (4)  give  the  coordinates  in  the  collinear 
configurations. 

Since  equations  (5)  have  the  same  form  as  the  differential  equations  in 
the  two-body  problem,  it  follows  that  in  the  collinear  solutions  the  orbits  are 
always  similar  conic  sections.  In  case  the  orbits  are  ellipses,  the  coefficient 
of  —  £/r3  and  —  rj/r3  is  the  product  of  the  cube  of  the  major  semi-axis  of  the 
orbit  and  the  square  of  the  mean  angular  speed  of  revolution.  If  the 
undetermined  scale  factor  be  so  chosen  that  x,  is  the  major  semi-axis  of  the 
orbit  of  mt  ,  the  mean  angular  velocity  of  revolution  of  the  system  is  w. 

The  hypothesis  is  made  that  or  and  ml  ,  .  .  .  ,  mn  are  real  positive 
numbers,  and  the  problem  is  to  find  the  number  of  real  solutions  of  (6)  for 
any  value  of  n.  For  each  of  these  solutions  there  is  a  six-fold  infinity  of 
collinear  configurations,  the  six  arbitraries  being  the  two  which  define  the 
plane  of  motion,  the  one  which  defines  the  orientation  of  the  orbits  in  their 
plane,  the  one  which  determines  the  epochs  at  which  the  bodies  pass  their 
apses,  the  one  which  determines  the  scale  of  the  system,  and,  finally,  the 
eccentricity  of  the  orbits. 

146.  Outline  of  the  Method  of  Solution.  —  The  method  of  solution 
involves  a  mathematical  induction  and  consists  of  the  following  steps: 

Assumption  (A).  It  is  assumed  that  for  n  =  v  the  number  of  real 
solutions  of  (6)  for  x^  ,  .  .  .  ,  xv  is  Nv  ,  whatever  real  positive  values  or 
and  m,  ,  .  .  .  ,  mv  may  have.  It  is  known  from  the  work  of  Lagrange  that 
when  v  —  3  the  number  is  Ns  =  3  =  f  3  !  . 

Theorem  (B).  If  to  the  system  w,  ,...,?«„  of  positive  masses  an 
infinitesimal  mass  mv+l  be  added,  then  the  whole  number  of  real  solutions 


Theorem  (C).  As  the  infinitesimal  mass  mp+l  increases  continuously 
to  any  finite  positive  value  whatever,  the  total  number  of  real  solutions 
remains  precisely  (v+l)Nv  . 

Conclusion  (D).  From  successive  applications  of  theorems  (B)  and 
(C)  it  follows  that  the  number  of  real  solutions  of  (6)  for  n  =  v+n  is 


Since  ^3  =  ^  3!  ,  it  follows  that  N3+lt  =  %  Gu-j-3)!  .     Let  /*+  3==n  and  we  have 

#.  =  }n!.  (7) 

To  complete  the  demonstration  of  this  conclusion  it  remains  only  to 
prove  theorems  (B)  and  (C). 


-I  K\K,H1-I,INK    Sill. I    |  KINS   FOR    H   BOD1KS.  L'S'.t 

147.  Proof  of  Theorem  (#). -\Yhcn  there  are  v  finite  bodies  m,,  .  .  .  ,  m, 

and  the  infinitesimal  body  «j,M  ,  equations  (6)  become 


M.r+l 

+0+ 


(8) 


r.r+l 


The  last  column  of  these  equations  is  zero  because  mr+1  =  0  (is  infini- 
tesimal). Consequently  the  first  v  equations,  which  involve  x,  ,  .  .  .  ,  xf 
alone  as  unknowns,  are  the  equations  defining  the  solutions  when  n  =  v. 
By  (A),  it  is  assumed  that  there  are  precisely  N,  real  solutions  of  these 
equations.  Let  any  one  of  these  solutions  be  x,  =  xj01,  .  .  .  ,  x,  =  a;™1. 
Then  the  last  equation  of  (8)  becomes 


t.H-1  *r+l  'r.r+I 


The  number  of  real  solutions  of  this  equation  is  required. 

Consider  <p,+\  as  a  function  of  z,+l  .    It  is  easily  verified  that 


l, v), 


(10) 


Since  y>r+1  is  finite  and  continuous  except  at  xf+l  =  xf\  .  .  .  ,  x*\  +00,  —  w, 
it  follows  that  there  is  an  odd  number  of  real  solutions  in  each  of  the 
intervals  -  oo  to  x™,  where  x™  is  the  smallest  x™,  xf  to  x{0>,  where 
x™1  and  x™  are  any  two  x(f  which  are  adjacent,  and  x*'  to  +00,  where 
x*  is  the  largest  x"'.  But  we  find  from  (9)  that 


ax 


r+l  'l.r+l 


which  is  negative  except  at  x,+1  =  x, ,  .  .  .  ,  x^^x, ,  where  it  is  infinite. 
Therefore  <p,+t  is  a  decreasing  monotonic  function  in  each  of  the  intervals, 
and  consequently  vanishes  once,  and  but  once,  in  each  of  them.  Since 


290  PERIODIC    ORBITS. 

there  are  >+l  of  these  intervals,  there  are,  for  each  real  solution  of  the 
first  v  equations  of  (8),  precisely  v  +  1  real  solutions  of  the  last  equation 
of  (8).  Since  the  first  v  equations  have,  by  hypothesis  (A),  Np  real  solu- 
tions, equations  (8)  altogether  have  precisely  (v-\-l)Nv  real  solutions. 
This  completes  the  demonstration  of  Theorem  (B). 


148.  Proof  of  Theorem  (C).— Let  Xj=xf  (j=l,  .  .  .  ,  v  +  l]  be  any 
one  of  the  (v-\-\)Nv  real  solutions  of  equations  (8)  which  are  known  to  exist 
when  mv+l  =  0.  It  will  be  shown  that  as  m,+l  increases  continuously  to  any 
finite  positive  quantity  whatever,  the  xf  can  be  made  to  change  continu- 
ously so  as  always  to  satisfy  equations  (8),  and  that  they  remain  distinct, 
finite,  and  real.  From  this  it  will  follow  that  there  are  at  least  (v-\-l)Nv 
real  solutions  of  (8)  for  every  set  of  finite  positive  values  of  ml ,  .  .  .  ,  mv+l . 
It  will  also  be  shown  that  no  new  solutions  can  appear  as  r»>+1  increases 
from  zero  to  any  finite  value.  Consequently,  it  will  follow  that  the  number 
of  real  solutions  of  (8)  is  exactly  (v  +  1)NV  for  all  finite  positive  values  of 
the  masses  m, ,  .  .  .  ,  mv+l  . 

The  roots  of  algebraic  equations  are  continuous  functions  of  the  coef- 
ficients of  the  equations  so  long  as  the  roots  are  finite  and  the  equations 
do  not  have  indeterminate  forms.  Consequently,  the  xf  are  continuous 
functions  of  mv+l  if  no  xf  becomes  infinite  and  if  no  xf  =  xf  .  The  real 
roots  of  algebraic  equations  having  real  coefficients  can  disappear  only  by 
passing  to  infinitjr,  or  by  an  even  number  of  real  solutions  becoming 
conjugate  complex  quantities  in  pairs.  Therefore  we  have  to  determine 
(1)  whether  any  finite  xf  can  become  equal  to  any  xf,  (2)  whether  any 
xf  can  become  infinite,  and  (3)  whether  any  two  real  solutions  can  become 
complex  for  any  finite  positive  values  of  ml  ,  .  .  .  ,  my+1 . 

(1).  The  masses  ml ,  .  .  .  ,  mv+l,  by  hypothesis,  are  all  positive.  Let 
the  notation  be  so  chosen  that  for  any  values  of  m1  ,  .  .  .  ,  mv+l  for  which 
the  xf  are  all  distinct  the  inequalities  xf  <xf  <  •  •  •  <  xf  <  z<°jj  are 
satisfied.  Suppose  that  as  some  mass  is  changed  the  difference  xf  -  xf 
approaches  zero  in  such  a  way  that  xf  and  xf  remain  finite ;  that  is,  rtj , 
which  occurs  only  in  the  expressions  <pt  and  <?_, ,  approaches  zero.  Suppose 
i  <  j.  Then  the  term  involving  rt)  becomes  negatively  infinite  in  <pt  and 
positively  infinite  in  <p} .  Consider  <pt  =  0.  Another  rik  must  approach  zero 
in  order  to  restore  the  finite  value  of  the  function  <pt ,  and  the  term 
involving  rik  must  become  positively  infinite  as  rtk  approaches  zero.  There- 
fore k<i.  But  r(k  enters  besides  only  in  <pt ,  and  similar  reasoning  shows 
that  rH  ,  where  I  <  k,  must  also  approach  zero.  In  this  manner  we  are 
driven  to  the  conclusion  finally  that  an  rw,  where  one  of  the  subscripts  is 
unity,  approaches  zero.  Then  consider  <p,  =  0.  All  its  terms  except  —  co2  x1 
are  negative,  and  since  one  of  its  rit ,  viz.  rft ,  approaches  zero  the  first 
equation  of  (8)  can  not  be  satisfied.  Consequently  the  original  assumption 


-i  u\ii,ii  I-I.IM:  >«>u  IIM\>  H»H  n  BODIES.  291 

that  some  r.  can  :i|)|>rii:ich  zero  for  finite  values  of  j-™',  .  .  .  ,  xJJ^.,  and 
finite  positive  values  of  w,  ....  m;  ,  leads  to  an  impossibility,  and  it  is 
therefore  false. 

(2).  On  multiplying  equations  (8)  by  m,  ,  w,,  .  .  .  ,  mr+1  ,  respectively, 
and  adding,  it  is  found  that 


It  follows  from  this  etiuation  that  no  x',01  alone  ran  become  infinite,  and  that 
if  one  of  them  becomes  negatively  infinite  then  some  other  one  must  become 
positively  infinite. 

Suppose  the  notation  is  again  so  chosen  that 


Then,  if  any  .r.  become-  negatively  infinite  .r,0)  must  also  become  negatively 
infinite,  and  from  the  equation  above  it  follows  that  x,+1  must  become 
positively  infinite.  Now  suppose  this  occurs  and  consider  ^  =  0.  In  order 
that  this  equation  may  remain  satisfied,  xJ0)  must  also  become  negatively 
infinite  in  such  a  way  that  xj"  —  x™'  shall  approach  zero.  But  now  it  follows 
from  <pt  =  Q,  since  -w'xj01  and  m,  (x{0)  -  xf)/i\  are  both  positive,  that  x™' 
must  also  become  negatively  infinite  in  such  a  way  that  x™'  ~xf  shall 
approach  zero.  Then  it  follows  similarly  from  <pt  =  0  that  x}0)  must  become 
negatively  infinite  in  such  a  way  that  xj0'  —  x{0>  shall  approach  zero.  This 
reasoning  continues  until  it  is  found  that  .rl°'  ,  .  .  .  ,  x™,  must  all  become 
negatively  infinite.  But  xj,°j,  at  least  must  become  positively  infinite. 
Therefore  x™'  can  not  become  negatively  infinite,  and  similarly  x™|,  can 
not  become  positively  infinite.  Hence  no  x',0'  can  become  infinite. 

In  order  to  prove  now  that,  as  m,+,  approaches  zero,  equations  (8) 
and  their  solutions  remain  always  determinate,  and  that  there  are  accord- 
ingly no  solutions  besides  those  obtained  in  theorem  (B),  consider  a  solution 
x,  ,  .  .  .  ,  xr+1  ,  in  which  the  x,  are  all  distinct  for  a  set  of  positive  values 
of  wi,  ,  .  .  .  ,  mr+i  ,  and  then  let  m,  approach  zero  as  a  limit. 

In  the  first  place,  if  x,  approaches  neither  x,+,  nor  x,_,  as  a  limit  as  m, 
approaches  zero  as  a  limit,  then  by  the  reasoning  of  (1)  and  (2)  above  no 
x,  can  approach  any  x»  as  a  limit. 

In  the  second  place,  x,  can  not  approach  x,+1  as  a  limit  as  m,  approaches 
zero  as  a  limit  unless  x,_i  approaches  xi+t  as  a  limit,  for  otherwise  <p,  =  0  can 
not  be  satisfied.  But  if  x,_,  approaches  x,+,  as  a  limit  as  m,  approaches  zero 
as  a  limit,  then  ^>,_,  =  0  and  <(>l+t  =0  can  not  be  satisfied  unless  x,_2  and  x,+, 
respectively  approach  x,  as  a  limit.  This  shifts  the  difficulty  to  v»(-»"0 
and  <pH_3  =  0,  and  so  on  until  <p,  =  0  and  tf>f+i  =0  are  reached,  which  can  not 
be  satisfied  under  the  hypotheses  it  has  been  necessary  to  make. 

In  the  third  place,  x,  can  not  become  positively  infinite  as  m,  approaches 
zero,  for  then  <(>,  =  Q  can  not  be  satisfied  unless  Xi_i  becomes  infinite  in  such 


292 


PERIODIC   ORBITS. 


a  way  that  x,  —  x(_i  approaches  zero.  Continuing  through  <p,_!  =  0,  .  .  . 
we  are  led  to  the  conclusion  finally  that  xl ,  .  .  .  ,  xp+l  all  become  posi- 
tively infinite,  but  then  the  center  of  gravity  equation  can  not  be  satisfied. 
Consequently  the  solutions  all  remain  regular  as  mt  approaches  zero  as 
a  limit. 

(3).  Since  the  solutions  of  (8)  are  continuous  functions  of  mv+1 ,  it 
follows  that  no  two  solutions  which  are  real  for  mv+1  =  0  can  ever  become 
conjugate  complex  solutions  for  any  real  value  of  mv+l  without  having  first 
become  equal;  and,  conversely,  no  two  solutions  which  are  complex  for 
mv+1  =  0  can  ever  become  real  for  any  real  value  of  mv+1  without  having  first 
become  equal.  Consequently,  if  a  multiple  solution  of  (8)  is  impossible  for 
every  set  of  finite  positive  values  of  raj ,  .  .  .  ,  mv+l ,  it  is  impossible  that 
any  real  solutions  should  disappear  by  becoming  complex,  or  that  any 
complex  solutions  should  become  real. 

The  conditions  that  x  =  xm  shall  be  a  multiple  solution  of  f(x)  =  0  are 
that  /(z(0))  =0  and  df  (x(w)/dx  =  0.  The  corresponding  conditions  that  a 
set  of  simultaneous  algebraic  equations  shall  have  a  multiple  solution  are 
that  a  set  of  values  of  the  variables  shall  satisfy  the  equations  and  that  the 
Jacobian  of  the  functions  with  respect  to  the  dependent  variables  shall 
vanish  for  the  same  set  of  values.  That  is,  the  conditions  that 


_(o> 

— 


,H-D, 


shall  be  a  multiple  solution  of  (8)  are  that  these  values  shall  satisfy  (8)  and 
also  the  equation 


dx, 


=  0. 


(11) 


Consider  two  solutions  of  a  set  of  algebraic  equations  having  real 
coefficients.  As  they  change  from  real  to  conjugate  complex  quantities,  or 
from  conjugate  complex  to  real  quantities,  for  some  value  of  a  continuously 
varying  parameter,  then  for  this  particular  value  of  the  parameter  they  are 
not  only  equal  but  they  are  real.  Consequently,  it  is  necessary  to  examine 
A  only  when  all  of  its  elements  are  real.  It  will  now  be  shown  that  it  can 
not  vanish  for  any  set  of  real  values  of  the  Xj  when  ml  ,  .  .  .  ,  mv+1  are 
positive,  and  consequently  that  it  can  not  vanish  for  any  particular  set 
which  satisfies  equations  (8).  When  this  is  established,  it  will  have  been 
proved  that  all  the  solutions  of  (8)  which  are  real  for  m(/+1  =  0  remain  real 
when  mv+l  increases  to  any  positive  value,  and  that  those  which  are  complex 
remain  complex. 


STRAKIHT-LINK    SOLUTIONS   FOK   Tl    BODIES. 

From  equations  (8)  and  (11),  it  follows  that 


I'm, 

f*~~   '      r3         » 

'  I.  r+l  '  J.  r+l 


where 


i      Mt   ,  .  .  .  , 


(12) 


A/,    =  -«'-    0    -  «p  -  • 

M*        2ttli  /\ 

,     =  -w' -j 0     -    • 
•  i* 


2m,,,  t 

M.  r+l 


2m, 


.r+l 


0. 


If  WH-,=  0  this  determinant  breaks  up  into  the  product  of  a  determinant  of 
the  same  type  as  (12)  and  a  factor  which  is  negative.  Therefore,  in  examining 
\\het  her  or  not  A  can  vanish,  it  is  sufficient  to  consider  the  general  case  in 
which  all  the  m,  are  positive. 

Several  properties  of  A  are  evident,  (a)  If  the  ilk  row  be  multiplied 
by  >nt  (i=l,  .  .  .  ,  v  +  1),  the  determinant  becomes  symmetrical.  (6)  The 
sum  of  the  elements  in  each  row  is  —  «*,  from  which  it  follows  that  the 
expansion  of  the  determinant  contains  w1  as  a  factor,  (c)  The  expansion 
«>f  the  determinant  contains  ( —  l)'+1o>1("+1>  as  one  of  its  terms,  and  since 
all  the  m,  are  positive  and  all  the  x,  are  real  the  sign  of  all  the  terms  coming 
from  the  product  of  the  elements  of  the  main  diagonal  is  (  — l)'+l. 

The  fact  is  that,  when  A  is  completely  expanded,  all  those  terms  not 
having  the  sign  (  — l)'+l  are  canceled  by  terms  coming  from  the  product  of 
the  main  diagonal  elements,  and  since  the  term  (-  l)r+V"+l)  is  certainly 
present  the  determinant  can  in  no  case  be  zero.  The  following  demon- 
stration of  this  fact  was  invented  in  1907  by  Dr.  T.  H.  Hildebrandt,  now 
of  the  University  of  Michigan,  as  a  class  exercise.* 

Since  the  determinant  contains  w'  as  a  factor,  every  term  in  its  expansion 
must  depend  upon  at  least  one  of  the  elements  of  the  main  diagonal.  Fasten 
the  attention  upon  any  term  of  the  expansion.  It  can  be  supposed  without 
loss  of  generality  that  it  depends  upon  the  first  main  diagonal  element.  In 
the  expansion  of  the  determinant  this  element  is  multiplied  by  its  minor; 
consequently  we  must  see  if  the  minor  can  vanish.  The  minor  is  of  the 

•An  earlier  proof  WM  devined  by  the  author,  and  rtffl  another  jointly  by  Prof««or  N.  B.  McLean,  of 
the  University  of  Manitoba,  and  Mr.  E.  J.  Moulton,  now  of  Northwestern  University. 


294  PERIODIC    ORBITS. 

same  form  as  the  original  determinant,  and  the  sum  of  the  elements  of  its 
ith  row  is  —  co2  —  m-i/rlf .  Consequently  every  term  in  the  expansion  of  the 
minor  which  does  not  vanish  will  contain  at  least  one  of  the  —  o>2—  m^/r^ 
as  a  factor.  But  these  elements  appear  only  in  the  main  diagonal  of  the 
minor.  Hence  all  terms  in  the  expansion  of  the  minor  which  do  not  vanish 
depend  upon  at  least  one  element  of  the  main  diagonal.  In  considering  our 
particular  term  it  may  be  supposed,  without  loss  of  generality,  to  depend 
upon  the  first  main  diagonal  element  of  the  minor.  In  the  expansion  of  the 
original  determinant  the  product  of  these  two  diagonal  elements  will  be 
multiplied  by  the  co-factor  of  the  minor  of  the  second  order  of  which  they 
are  the  main  diagonal.  This  co-factor  has  the  same  properties  as  the  first 
minor  just  considered,  and  in  the  same  way  it  is  proved  that  at  least  one  of 
its  diagonal  elements  must  be  involved  in  the  term  in  question;  that  is, 
the  term  under  consideration  depends  upon  at  least  three  elements  of  the 
main  diagonal.  On  continuing  in  this  manner  it  is  proved  that  any  term 
in  the  final  expansion  depends  upon  all  the  elements  of  the  main  diagonal, 
which  are  all  of  the  same  sign  in  every  one  of  their  terms.  Consequently, 
all  the  terms  which  do  not  cancel  out  in  the  expansion  of  the  determinant 
have  the  sign  (  —  1)"+1;  and  it  has  been  seen  that  there  is  at  least  one  such 
term,  viz.  ( —  1)"+1  co2<"+1).  Therefore  the  determinant  not  only  can  never 
vanish,  but  it  can  never  be  less  than  a>2'"+u  in  numerical  value. 

Since  A  can  never  vanish  for  real  distinct  x(f  when  all  the  m,  are  real 
and  either  zero  or  positive,  it  follows  that  no  real  solutions  can  ever  be  lost 
or  gained  as  the  m,  vary  so  as  not  to  become  negative,  and  therefore  that 
the  number  of  real  solutions  of  (8)  is  (v+\}Nv  =  %(v+\}\  for  all  positive 
finite  values  of  ml ,  .  .  .  ,  ?nv+l ,  and  co2. 

149.  Computation  of  the  Solutions  of  Equations  (6). — There  are  well- 
known  methods  of  finding  the  roots  of  a  single  numerical  algebraic  equation 
of  high  degree,  but  they  are  not  readily  applicable  to  simultaneous  equations 
of  high  degree.  However,  when  the  order  of  the  masses  has  been  chosen, 
equations  (6)  will  become  polynomials  in  x{  ,  .  .  .  ,  xn  after  they  have  been 
cleared  of  fractions.  Then  by  rational  processes  n—  1  of  the  xt  can  be 
eliminated  from  these  equations,  giving  a  single  equation  in  the  remaining 
unknown.  The  solutions  of  this  equation  can  be  found  by  the  usual 
methods  and  the  results  can  be  used  to  eliminate  one  unknown.  By  repeated 
application  of  this  process  to  the  successively  reduced  equations,  the  solu- 
tions can  all  be  found.  The  one  satisfying  the  conditions  of  reality  of 
xl  ,  .  .  .  ,  xn  and  their  order  relation  is  the  one  desired. 

The  solutions  of  (6)  can  also  be  found  by  a  method  closely  related  to 
that  by  means  of  which  their  existence  was  proved  above.  Suppose  for 
mt  =  mf  (i=l,  .  .  .  ,  ri)  a  solution  xf  =  xf  of  equations  (6)  is  known. 
The  m f  are  supposed  to  be  zero  or  positive.  Suppose  it  is  desired  to  find 
the  corresponding  solution,  that  is,  the  one  in  which  the  masses  are  arranged 


8TKAICHT-LINE   SOLUTIONS   FOR    n    BODIES.  J'.t.') 


on  the  line  in  the  same  order,  for  m  =///,+/«,.  Let  the  corresponding 
set  of  the  *,  satisfying  «>)  be  .r,  =  or',01  +  {, ,  where  the  £,  are  functions  of 
M,,  .  .  .  ,M.  to  be  determined.  On  substituting*,  =  .rr  +  £<  and  mt  =  m™+n( 
in  (6),  making  use  of  the  notation  of  (8),  expanding  tus  power  series  in  the 
{,  and  /j,  (which  is  always  po»ihle.  since  it  has  been  shown  that  no  a:',0'  can 
become  infinite  and  no  .< •,'  can  e(|iial  any  .r'"'),  and  remembering  that  x,  =  x(?' 
-olution  of  (6)  for  m,  =  ///"''  ,  the  resulting  equations  are  found  to  be 


(13) 


S<L«?>  1  4.  v  j  r  T  ^  *  T  -  v  d*>  u 

h£"L£a*ij  2/a^/ 

S^!t_i_y>J_rv^.tT-  y  d*>*  u 

,_,«*/     ^ZalZdx/  £««/ 


where  the 


are  the  symbolic  powers  used  in  connection  with  the  power-series  expan- 
sions of  functions  of  several  variables. 

The  determinant  of  the  terms  of  the  first  degree  in  the  %,  in  equations 
i::  is  the  ^  of  equation  (11),  which  has  been  proved  to  be  distinct  from 
zero  in  this  problem.  Therefore  equations  (13)  can  be  solved  by  the  method 
explained  in  §1,  and  by  §2  the  solutions  converge  for  |/x,|>0,  but  sufficiently 
small.  Suppose  they  converge  if  |M<|^r.  Keeping  the  MI  within  this  limit, 
a  solution  xt  =  x\"  is  computed.  Then  this  can  be  used  as  a  starting-point 
for  a  second  application  of  the  process,  which  can  be  related  as  many  times 
as  may  be  desired. 

Hence,  to  find  the  solution  in  which  the  bodies  «,,...,  inm  have  any 
finite  positive  values  and  lie  in  a  determined  order  on  the  line,  we  may  start 
with  /«,  ,  m,  ,  and  m,  and  solve  the  Lagrangian  quintic*  which  defines  their 
distribution  on  the  line.  Then  an  infinitesimal  body  mt  is  added  and  its 
position  is  found  by  solving  the  single  equation  (9),  in  which  v  =  3.  This 
infinitesimal  mass  mt  is  made  to  increase,  step  by  step,  to  the  required  finite 
value  and  the  corresponding  values  of  z,  ,  .  .  .  ,  z4  are  computed.  It 
follows  from  the  fact  that  the  d<pi/dx,  are  less  than  fixed  finite  quantities 
depending  upon  w!  and  m,  ,  .  .  .  ,  m,  ,  while  A  is  not  less  than  «"  in  numerical 
value,  that  any  finite  value  of  m4  can  be  reached  in  this  way  by  a  finite 
number  of  steps.  After  the  required  value  of  m«  has  been  reached,  the 
process  can  be  repeated  for  /«„  etc.,  to  any  finite  number  of  bodies.  Not- 
withstanding the  fact  that  this  would  be  very  laborious  if  the  number  of 
bodies  were  large,  we  must  regard  the  problem  as  completely  solved  both 
theoretically  and  practically. 

•Ttaewnd's  \ttcaniquc  Crlette,  vol.  I,  p.  155,  or  Moulton'a  Introduction  to  CeleHial  Meclianiet,  p.  216. 


296 


PERIODIC    ORBITS. 


II.    DETERMINATION  OF  THE  MASSES  WHEN  THE 
POSITIONS  ARE  GIVEN. 

150.  Determination  of  the  Masses  when  n  is  Even.— Suppose  w  and 
the  n  distinct  points  on  a  line,  xt ,  .  .  .  ,  xn ,  are  given,  and  consider  the 
problem  of  determining  ml ,  .  .  .  ,  mn  so  that  the  circular  solutions  shall 
exist.  There  will  be  no  loss  of  generality  in  selecting  the  notation  so  that 
Xi<x,<  •  •  •  <xn.  With  this  choice  of  notation  equations  (6),  which  are 
necessarj^  and  sufficient  conditions  for  the  solutions,  become 


T     W      T-3      '1 

'12 

2      i             T3      -r   ^ 

13                                      'l.n-1             'In 
*»                                 1     Win-1     i        f^n 

+  pr  r    p-  - 

-  ar«j 

_  /,i2a; 

r1  r2 

'l,n-l          '2,n-l 


0 


r2 

'In 


'  B,n-l 


(14) 


The  m4  enter  these  equations  linearly  and  are  therefore  uniquely  deter- 
mined if  the  determinant 


7)  = 


i     n 

,      1 

1     ^ 

+                            1        •'• 

T     U    , 
1 

»n  ' 

^*13 

1 

y.*            J                   y* 
1           J                              J 

r2    ' 
'12 

+    ,.2      >      •     •     •     > 

T  3        ,       +  r2 

'2,n-l                     '2n 

1 

1 

1 

1 

»to  ' 

r-jn  ' 

r2      ,      •     •     •     , 
'3n 

r.2            ,         + 

(15) 


is  distinct  from  zero.  This  is  a  skew-symmetrical  determinant,  and  when 
n  is  even  it  is  the  square  of  an  associated  Pfaffian,  and  therefore  is  not  in 
general  zero.  Therefore  if  n  is  even  the  masses  are,  in  general,  uniquely 
determined  when  «  and  xl ,  ,  .  .  ,  zn  are  given,  though  it  should  be  noted 
that  they  are  not  necessarily  all  positive. 

151.  Determination  of  the  Masses  when  n  is  Odd. — In  this  case  the 
skew-symmetric  determinant  is  identically  zero,  but  its  first  minors  of 
the  main  diagonal  elements,  being  skew-symmetrical  determinants  of  even 
order,  are  in  general  all  distinct  from  zero ;  consequently  the  xt  must  satisfy 
one  relation  in  order  that  equations  (14)  shall  be  consistent.  To  get 
this  relation,  take  the  right  members  to  the  left  and  add  the  equation 


STRAIGHT-LINE   SOLUTIONS  FOR   n   BODIES. 


297 


rn,  x,  +  ?»,  x,+  •  •  •  +  w.  x,  =  0,  which  is  a  consequence  of  (14),  to  the  set 
of  equations.     There   are   then   n  +  1    linear   homogeneous  equations   in 


—  w    7« 


m, 


In  order  that  they  shall  be  consistent  their  eliminant 
0  ,     +  x,  ,     +  x, ,...,+  x. 


I    • 


0,     +i 
*« 

1 


~Xu,        ~  -»~  »        ~  .f  ,    •    • 
'in  'In 


>!. 


0 


must  vanish.    This  is  also  a  skew-symmetrical  determinant  and  is  the  square 
of  the  Pfaffian  F,  where 


F=|*,,     x,,  .  .  .,    *. 
l  1 


'•-I.* 


(16) 


Equation  (16)  can  be  found  also  by  solving  any  n— 1  equations  of  (14) 
for  the  corresponding  m,  and  substituting  the  solutions  in  the  remaining 
one.  The  result  is  a  sum  of  determinants  which  can  be  shown  to  be  the 
expansion  of  F  multiplied  by  the  square  root  of  the  determinant  of  the 
coefficients  of  the  n  —  1  masses  m,  in  the  equations  used. 

When  F  =  0  is  satisfied  by  x, ,  .  .  .  ,  x.,  equations  (14)  are  consistent. 
Then,  after  any  w,  has  been  chosen  arbitrarily,  the  corresponding  n—l 
equations  can  in  general  be  uniquely  solved  for  the  remaining  m, ,  and  the 
unused  equation  will  be  satisfied  because  F  =  Q. 

1 52.  Discussion  of  Case  n  =  3. — When  n  =  3  the  determinant  D  becomes 


and  the  Pfaffian  F  is 


(ruraru)' 


(17) 


It  will  now  be  shown  that  when  any  two  of  x, ,  x, ,  x,  are  so  chosen  as 
to  satisfy  the  conditions  x,<x,  <x, ,  the  third  is  uniquely  determined  by 
(17)  and  these  inequalities.  From  the  fact  that  in  this  case  ru>ra,  ru>ru, 
it  follows  that  if  x,  is  positive,  then  —  xt/r*a+xl/rtll  is  positive,  and  therefore 
that  x,  must  be  negative  in  order  that  (17)  may  be  satisfied.  If  x,  is 
negative,  x, ,  being  less,  must  also  be  negative.  That  is,  x,  is  necessarily 
negative;  and  similarly  x,  is  necessarily  positive. 


298  PERIODIC    ORBITS. 

Suppose  x2  and  x3  are  chosen  and  consider  F  as  a  function  of  xl  .     Then 
it  follows  at  once  that 


From  the  inequalities  x2<x3  and  r12<rM  ,  it  follows  that  dF/dx±  is  positive 
for  —oo  <x1<x2.  Therefore  there  is  but  one  solution  of  (17)  for  x1<xi 
when  x2  and  a  positive  x3  are  chosen.  By  symmetry,  there  is  but  one 
solution  of  (17)  for  x3>x2  when  x2  and  a  negative  xl  are  chosen. 

If  a;,  is  negative  and  x,  is  positive,  but  both  otherwise  arbitrary,  F 
considered  as  a  function  of  x2  gives 


<JX2  ?23  ^*13  M2 

Therefore  there  is  but  one  solution  of  equation  (17)  for  x.2  which  satisfies 
the  inequalities  xl  <  x2  <  x3 . 

Suppose  a  negative  xl ,  a  positive  x3 ,  and  mz  are  given  arbitrarily  and 
that  z2  is  defined  by  (17).     Then  equations  (14)  give 

(20) 

If  ra2  is  negative,  mt  and  m3  are  necessarily  positive.  If  m2  is  positive  and 
sufficiently  small,  both  rn,  and  m3  are  positive.  As  m2  increases,  inl  and  m3 
decrease.  Suppose  x3>  —  xt .  Then,  for  a  certain  positive  value  of  w2  the 
mass  m3  vanishes  while  ra2  is  still  positive.  For  a  certain  greater  value  of 
mj,  the  mass  m1  is  zero  and  m,  is  negative.  For  still  greater  values  of  w2 , 
both  m1  and  m3  are  negative.  From  the  fact  that  xt  must  be  negative 
and  x3  positive,  and  from  equations  (20),  it  follows  that  not  all  three  of 
the  masses  m1}  mt,  and  ms  can  be  negative  simultaneously. 


CHAPTER  IX. 

OSCILLATING  SATELLITES  NEAR  THE  LAGRANGIAN 
EQUILATERAL-TRIANGLE  POINTS. 


By  THOMAS  BUCK. 

153.  Introduction.  —  This  chapter  is  devoted  to  an  investigation  of 
certain  periodic  orbits  which  an  infinitesimal  body  may  describe  when 
attracted  according  to  the  Newtonian  law  by  two  finite  bodies  revolving  in 
circles  about  their  center  of  mass.     It  has  been  shown  by  Lagrange  that 
three  bodies  placed  at  the  vertices  of  an  equilateral  triangle  can  be  given 
such  initial  projections  that  they  will  retain  always  the  same  configuration. 
The  orbits  here  considered  are  in  the  vicinity  of  the  equilateral-triangle 
points  defined  by  the  two  finite  bodies.     The  infinitesimal  body  is  displaced 
from  the  vertex  of  the  equilateral  triangle,  and  its  initial  projection  is  deter- 
mined so  that  its  motion  is  periodic  with  respect  to  that  of  the  finite  bodies. 
The  existence  of  the  solution  is  established  by  the  method  of  analytical 
continuation.     The  construction  is  made  by  the  method  of  undetermined 
coefficients,  using  the  properties  obtained  in  the  discussion  of  the  existence. 
The  solutions  are  given  in  the  form  of  power  series  which  converge  for 
sufficiently  small  values  of  the  parameter  employed. 

154.  The  Differential  Equations.  —  The  motion  of  the  infinitesimal  body 
will  be  referred  to  a  rotating  system  of  axes,  the  origin  being  at  the  center  of 
mass,  the  £»j-plane  being  the  plane  of  the  motion  of  the  finite  bodies,  and  the 
rate  of  rotation  such  that  they  remain  on  the  £-axis.     The  masses  of  the  finite 
bodies  will  be  represented  by  n  and  1  —  n  so  taken  that  n  ^$  ,  their  distance 
apart  will  be  taken  as  the  unit  of  distance,  and  the  unit  of  time  will  be 
so  chosen  that  the  proportionality  factor  k?  is  unity.     Then  the  equations 
of  motion  for  the  infinitesimal  body  are 


where 


_  ___  -. 

I       #    #  '        de      dt  "  dr,  '        de  '  ar  ' 


r,  and  r,  being  the  distances  from  the  infinitesimal  body  to  the  bodies  1  —  j 
and  M  respectively. 


300  PERIODIC    ORBITS. 

The  Lagrangian  equilateral  triangle-solutions  are 


ii.  IO=-M,        »?o=-V3,        r»=o. 

The  two  points  in  the  rotating  plane  defined  by  these  solutions  will  be 
referred  to  as  point  I  and  point  II  respectively.  The  question  of  the  exist- 
ence of  periodic  solutions  of  equations  (1)  in  the  vicinity  of  these  points  is 
to  be  investigated.  For  this  purpose  the  origin  is  transferred  to  the  point 
in  question  by  means  of  the  transformation 

|=£-M+z,  v=±^V3+y,  r=*.  (2) 

After  the  transformation  is  made  the  right  members  of  the  equations  are 
expanded  as  power  series  in  x,  y,  and  z.  The  region  of  convergence  of 
these  series  is  determined  by  the  singularities  of  the  functions  l/rl  and  l/r2  . 
When  point  I  is  considered,  the  region  of  convergence  is  given  by  the  values 
of  x,  y,  and  z  satisfying  the  inequalities 


This  region  consists  of  the  common  portion  of  two  spheres,  excluding  their 
centers  which  are  at  the  finite  bodies,  each  of  radius  \/2.  For  point  II  the 
region  of  convergence  is  defined  by  the  inequalities 

-l<:c2+7/2+22-Hc-\/32/<+l,  -l<z2-H/+z2-.r-V37/<-fl. 

Since  the  origin  in  this  case  is  at  the  second  point  it  follows  that  this  region 
is  the  same  as  that  found  for  the  first  point. 

As  the  two  cases  differ  only  in  the  sign  before  the  vl5,  it  is  necessary 
to  consider  in  detail  only  one  of  them.  The  discussion  will  be  given  for 
point  I  with  the  understanding  that  by  changing  the  sign  of  \/3  the  corre- 
sponding expressions  for  the  point  II  are  obtained. 

Two  parameters,  e  and  8,  are  introduced  as  in  Chapter  V.  Then, 
denoting  derivation  as  to  r  by  accents,  the  differential  equations  become 


(3) 


ox  1I.I.AI1.V,    >\u;i.U||->,    Kgi  11.  \1KHAL-TKIANGLE    CASE.  301 

where 


A",  =  +  i  [-  37z«+75v/3(l  - 


(4) 


For  sufficiently  small  values  of  x,  y,  z,  and  e  these  series  are  all  convergent. 
Equations  (1)  admit  the  integral 


The  corresponding  integral  of  equations  (3)  can  be  expressed  as  a  power 
series  in  «.    The  terms  independent  of  e  are 


(5') 
These  terms  will  be  found  useful  in  the  existence  proofs  which  follow. 

155.  The  Characteristic  Exponents.  —  For  «  =  6  =  0  equations  (3)  become 


(6) 


The  last  equation,  being  independent  of  the  first  two,  can  be  integrated 
immediately,  giving 


302  PERIODIC    ORBITS. 

To  integrate  the  first  two  equations,  let 


On  substituting  in  the  first  two  equations  of  (6)  and  dividing  out  eXT,  we  have 


(7) 


[2X- 


In  order  that  these  equations  may  be  satisfied  by  values  of  K  and  L  different 
from  zero,  the  determinant  of  the  system  must  vanish.  This  gives  for  the 
determination  of  X  the  characteristic  equation 

X4+X2+fM(l-M)=0.  (8) 

Each  of  the  four  values  of  X  satisfying  this  equation  gives  a  particular  solu- 
tion of  equations  (6).  The  corresponding  K  and  L  must  satisfy  equations 
(7).  Since  these  equations  have  a  vanishing  determinant  the  ratio  only  of 
the  K  and  L  is  determined.  In  what  follows  K  will  be  considered  as  arbi- 
trary, and  L  will  be  determined  in  the  form  L  =  bK. 

In  order  that  a  solution  shall  be  periodic,  the  corresponding  X  must  be 
a  purely  imaginary  quantity.     Upon  solving  (8),  we  have 


For  small  values  of  /*  the  roots  of  (8)  are  pure  imaginaries;  the  limiting 
value  of  M  for  which  this  is  true  is  given  by  the  equation 

1-27M(1-M)=0. 

The  root  of  this  equation  which  is  less  than  \  is  /JL  =  .0385  ....  For 
fj.  ^  .0385  .  .  .  the  values  of  X  are  purely  imaginary  and  the  corresponding 
particular  solutions  are  periodic.  Let  <rl  and  <r2  be  two  numbers  defined  by 


2_  l  +  Vl-27M(l-M)  ,      l-Vl-27M(l-M) 

ffl~  2  2" 

It  follows  that  ar1  and  o-2  do  not  exceed  unity  for  n  ^  .0385  .  .  .  ,  and  that 
cr^ov,.  Then  the  roots  of  (8),  which  are  the  characteristic  exponents  of 
the  problem,  become  ±o-iV  — l  and  ±<r2V—  1- 


OSCILLATINC.    SATKI.I.l  I  !•>.    KqUILATEKAL-TKIAM.I.K    CASE. 

156.  The  Generating  Solutions.—  The  general  solution  of  (6)  is 

r=  ,/,,."»  -"4.  a,e-"v->T+  o,e"v'-lr+  a4c-"v-'r, 
ij  =  b,  «,  <•'-  '  r  +6,  a,  e~"»  •'-'  r+  6,  a,  e"^~lT+64  a,  c~"  *-'  ', 
2=c,sinT-r-c,cosT. 

The  quantities  ",,«,,  a,  ,  o4  ,  c,  ,  and  c,  are  arbitrary,  while  6,  ,  6,  ,  6,  ,  and 
6,  are  determined  by  equations  (7)  when  the  proper  values  of  X  are  sub- 
stituted. Thus  it  is  found  that 

h  - 


h.      -<,->-M  . 


Various  periodic  solutions  are  obtained  from  this  general  solution  by  assign- 
ing suitable  values  to  the  arbitrary  constants  and  to  the  quantity  p.  For 
M<.0385  ...  we  have  two  distinct  periodic  solutions: 


y  =  blal  c"v/-'r+  6,0,6-"^'-'  T,     2  =  0. 
II.    x 


These  equations  represent  ellipses  in  the  .n/-planc  with  centers  at  the  origin. 
The  major  axes  of  the  ellipses  coincide  and  make  an  angle  0  with  the  positive 
z-axis  defined  by 

tan20=->/3(l-2M), 

with  ros20  positive.  The  major  and  minor  axes  of  the  second  are  greater 
and  less  respectively  than  those  of  the  first.  The  periods  are  1-K/a^  and 
2jr/<r,  respectively.  If  n  =  .0385  ...  it  follows  that  a,  =  <r,  ,  and  solutions 
I  and  II  coincide. 

For  all  values  of  ju  we  have  the  periodic  solution 

III.  x  =  0,         y  =  0,         2  = 

This  solution  defines  an  oscillation  on  the  z-axis  with  the  period  2r. 

It  is  possible  to  give  n  values  such  that  <r,  and  <TJ  are  relatively  commen- 
surable. Let  mtffl  =  mlff.  ,  where  /«,  and  w,  are  integers.  Then,  by  using 
the  definitions  of  a,  and  <r,  ,  we  find 


304 


PERIODIC    ORBITS. 


For  n  <  .0385  .  .  .  the  expression  on  the  left  takes  all  values  on  the  interval 
from  0  to  1.  By  choosing  w,  and  w2  so  that  0<  (ml  —  m%)/(m\-\-wQ  <  1,  and 
solving  the  equation  for  n,  we  have  a  value  of  ^  making  al  and  <r2  com- 
mensurable. For  such  values  of  M  we  have  the  additional  periodic  solution 


IV. 


x  = 


z=0, 


~ffl  v~l  T 
~'1  v=1  T 


the  period  of  which  is  2miTr/<Ti  = 

By  an  argument  precisely  similar  to  the  preceding  it  can  be  shown 
that  for  special  values  of  n,  the  characteristic  exponents  o-t  and  <r2  separately 
may  be  commensurable  with  unity.  We  have  then  the  periodic  solutions 


V. 


x= 


=Ci  sin  r+c2  cos  T; 


x=     a, 


y  =  63  a3  enV~1T  +64  a4 
z  =c1sinr+c2cosr. 


VI. 


The  periods  are  2nnr  =  2m1w/al  and  2nw  =  2n2ir/<r,,  respectively. 

Finally,  o-j  and  o-2  may  be  commensurable  with  unity  for  the  same 
value  of  /i. 

Let  a^  =  b  and  Cffi  =  d;  then 

262-a2      c2 


or  c' 

From  this  relation  it  follows  that  a2d2  =  c2(a2  —  62).  If  now  we  give  a,  b,  c, 
and  d  such  integral  values  that  this  relation  is  satisfied  and  such  that 
(2fe2  — a2)/a2  lies  between  0  and  1,  the  corresponding  value  of  n  will  make  <rl 
and  0-2  commensurable  with  unity.  For  example,  such  a  choice  is 

a  =  c=13,         6=12,         d  =  5. 
For  such  values  of  M  we  have  the  periodic  solution 


VII. 


x= 


The  period  of  this  solution  is 


OtdLLATOfG    >\  II.I.I.I  I  l>.    KijULATERAL-TUIANGLE    CASE. 

Those  periodic  -ohition.-  are  tin-  generating  solutions  for  the  general 
problem.  \Ye  sh;ill  now  suppose  tlial  «  is  not  zero  and  collider  the  question 
of  the  existence  of  the  continuations  of  these  .solutions  with  re>|)e<-t  to  the 
parameter  t.  The  period  in  the  variable  r  will  in  all  cases  be  taken  the  same 
as  that  of  the  generating  solution.  The  |x>riod  in  t  of  the  solution  is  found 
from  the  relation  l=(l  +  6)r. 


157.  General  Periodicity  Equations. — For  «  =  0  the  general  solution 

of  equations  uli  is 


x= 


2  =r,sn 


(9) 


Normal  \ariables  are  introduced  by  the  transformation 


x  =  ^1+9  xl 

y'  =  a,  (  1  +  *)  v^T  (6,  x,  -  bt  xt)  +  at  (  1+  i)  v^T  (6, 

The  differential  equations  then  become 


(10) 


(11) 


where 


A  being  the  determinant  of  the  transformation  (10),  and  A,,  the  minor  of  an 
element  in  this  determinant.  The  first  subscript  indicates  the  row  and  the 
second  one  the  column. 


306  PERIODIC    ORBITS. 

For  €  =  5  =  0  the  initial  values  of  the  variables  xt  ,  x2  ,  x3  ,  and  z4  are 
flj  ,  a2  ,  a,  ,  and  a4  respectively.  In  the  general  problem  we  take  as  the  initial 
conditions 


Since  there  is  a  component  of  force  always  directed  toward  the  xy-plane,  it 
is  clear  that  at  some  time  z  must  be  zero.  Hence  we  have  supposed  that 
z  =  0at  r  =  0. 

According  to  §§14  and  15  equations  (11)  can  be  integrated  as  power 
series  in  the  parameters  alt  Oj,  a3,  a4,  7,  5,  and  e,  which  converge  for 
|a,|  ,  .  .  .  ,  |  e|  sufficiently  small,  and  for  O^r^T,  the  value  of  T,  which  in 
this  case  is  the  period,  being  given  in  advance.  These  solutions  have  the 
form 

31=(a1+a1)e+<na+8)^~lT+€p,  (at,  a,,  a,,  o4f  7,  6,  e;  T), 
z,=  (a,+a,)  e-^'+'^'  +  ep,  (a,,  a,,  a3,  o4,  7,  5,  e;  T), 
a;3  =  (a,+a3)  e+"<1+*>^'+ep3  (aM  a,,  o,,  a4,  7,  6,  e;  r), 
Z4=(a4+a4)  e-x'+'^'+ep,  (a,,  a,,  o,,  o4,  7,  5,  ej  T), 
2  =(C+T)  sin(l  +  S)T+ep,(a,,  a2  ,  a3  ,  a4  ,  7,  5,  e;  T), 
Z'  =  (l  +  3)(c+7)cos(l+5)r+ep6(a1,  a2  ,  a,  ,  a4  ,  7,  5,  e;  T). 

The  general  periodicity  equations  for  the  period  T  are 
s.(0)  =  0  (t  =  l,  ...  ,4), 


(13) 


z(T)-  «(0)=0,  '/ 


These  equations  are  sufficient  conditions  for  the  periodicity  of  the  solution. 
On  solving  them  for  the  arbitraries  a,  ,  cu,  ,  a3  ,  a4  ,  7,  5  as  power  series  in  e, 
a  determination  of  these  quantities  is  obtained  such  that  the  corresponding 
solution  is  periodic.  Hence,  on  substituting  these  series  in  (13),  the  resulting 
expressions  for  the  xt,  z,  and  z'  are  periodic.  These  expressions  can  be 
rearranged  as  power  series  in  e  which  will  converge  for  e  sufficiently  small, 
and  for  all  0  ^  T  <!  T,  provided  the  values  of  a,  ,  .  .  .  ,  a4  ,  7,  8  obtained 
from  (14)  lie  in  the  domain  of  convergence  of  (13).  The  convergence  can 
be  secured  by  imposing  the  condition  that  the  expressions  for  the  arbitraries 
obtained  from  (14)  shall  be  power  series  in  e,  which  vanish  with  e.  The 
solutions  are  then  analytical  continuations  of  the  generating  solutions. 

The  periodicity  equations  will  now  be  set  up  for  each  of  the  generating 
solutions,  and  the  possibility  of  solving  them  for  the  arbitraries  in  the 
required  form  will  be  considered.  The  equations  wih1  be  written  for  the 
point  I  only.  The  conclusions  are  the  same  in  all  cases  for  the  point  II. 


MM  II.I.ATIN..    -\1K1.I.IIK-.    Kijl  II.  VIKKAL-TKI  ANGLE    CASE.  307 

158.  The  First  Generating  Solution.     The  explicit  form  of  equations 
1  I    i-  iiuu  to  IK-  determined  for  r  =  2jr/V1  and  <i,  =  a,  =  c  =  0.     On  account 
of  the  existence  of  the  integral  (5),  one  of  the  equations  is  redundant.     For 
if  we  let 


z'  =  Q+w', 

where  »/,(0)  =  «/,(0)  =  j/,(0)  =y,(0)  =  tp(0)  =M?'(0)  =0,  we  find  that  the  partial 
derivative  of  the  integral  (ii')  with  respect  to  y,  is 


for  r  =  2T/0-,  ,  and  //,  =  yt  =  y,  =  yt  =  w  =  w'  =  0.  The  integral  can  therefore  be 
solved  uniquely  for  //.  in  terms  of  y,  ,  yl  ,  y,,  w,  and  w'.  If  the  latter 
quantities  are  periodic  it  follows  that  y,  also  is  periodic.  Therefore  the 
>econd  equation  is  redundant  and  can  be  suppressed.  On  computing  the 
necessary  terms  of  (13),  it  is  found  that  the  remaining  equations  have  the 
form 


'] 


0, 


K—  t*n*/—i         -  ~i  r  ~i 

e—*—-l)+  •  •  •\+<[ella,+enai+ent+  •  •  -J=0, 

y   (     sin—       )  -f  •  •  '!+*! 


]=o, 


(15) 


where  the  explicit  computation  shows  that  the  et,  are  constants  different 
from  zero. 

The  right  member  of  the  2-equation  in  (1  1)  carries  the  factor  z.  Conse- 
quently the  solution  carries  the  factor  y,  and  hence  the  last  two  equations 
of  (15)  have  y  as  a  factor.  If  y  is  not  zero  and  is  divided  out,  there 
remains  a  term  in  each  equation  which  is  independent  of  the  arbitraries. 
These  terms  can  vanish  only  if  <r,  is  the  reciprocal  of  an  integer.  If  they 
do  not  vanish,  it  follows  that  the  equations  can  not  be  satisfied  by  the 
vanishing  of  all  the  arbitraries,  and  consequently  that  solutions  of  the 
required  form  do  not  exist. 

In  order  to  satisfy  equations  (15)  we  must  suppose,  then,  that  7=0. 
Hence  the  motion  of  the  infinitesimal  body  will  be  entirely  in  the  :rz/-plane. 
The  first  three  equations  are  satisfied  by  o,  =  a4  =  «=«  =0,  and  are  not  satis- 
fied by  o,  =  o,  =  5  =  0.  The  coefficient  of  6  in  the  first  equation  is  distinct 


308  PERIODIC    ORBITS. 

from  zero.  If  the  coefficients  of  a3  and  a4  in  the  second  and  third  equations 
respectively  are  also  distinct  from  zero,  it  follows  that  there  is  a  unique 
solution  for  a3  ,  a4  ,  and  5  as  power  series  in  e,  vanishing  with  e.  It  is  neces- 
sary, therefore,  that  vt  be  such  that  the  expressions 

27T 


e    "l     —1,        e     '»      —  1.         sin  —  .         cos  --  1 

al  <TI 

shall  not  vanish.  The  first  two  vanish  if  a-,  =  ma-1  ,  where  m  is  an  integer. 
Since  0-,  and  o-2  are  positive  and  o^  ^  o-2  ,  this  occurs  only  when  al  =  a2  . 
This  case  will  be  treated  later  in  the  discussion  of  the  commensurable 
cases.  It  will  be  shown  that  orbits  exist  in  this  case  also.  The  last  two 
expressions  vanish  only  if  ^  is  the  reciprocal  of  an  integer.  Suppose,  then, 
that  ffl  —  1/m.  On  solving  the  first  three  equations  of  (15)  for  a3,  a4,  6 
and  substituting  in  the  last  two,  there  remains,  after  dividing  out  e2  and  7, 
a  term  in  each  independent  of  the  arbitraries.  There  can  be,  then,  no  solution 
of  these  equations  in  the  required  form.  They  can  be  satisfied  only  by 
putting  7  =  0  as  before.  Hence  equations  (15)  have  a  unique  solution  of 
the  same  form  in  this  case  also. 

The  question  of  the  existence  of  an  orbit  re-entering  after  m  revolu- 
tions will  now  be  considered.  The  period  in  this  case  is  2m-n-/<rl  .  The 
periodicity  equations  have  a  unique  solution,  as  before,  except  when  n  is 
such  that  ?no-2  =  <r1.  In  this  case  the  second  and  third  equations  do  not 
admit  solutions  for  a3  and  a4  .  This  case  will  be  treated  later  in  the  discus- 
sion of  the  commensurable  cases.  It  will  be  shown  that  orbits  exist  for 
these  values  of  M  also.  Hence  there  is  a  single  orbit  re-entering  after  m 
revolutions.  But  one  such  orbit  is  obtained  by  m  repetitions  of  the  orbits 
re-entering  after  one  revolution.  It  follows,  therefore,  that  this  orbit  is  the 
only  one. 

The  periodicity  equations  have  now  been  satisfied  with  at  and  a2  still 
remaining  arbitrary.  Since  we  now  have  z  =  0,  one  relation  between  these 
arbitraries  is  obtained  by  fixing  the  origin  of  time.  It  will  be  supposed 
that  x'  =  0  at  r  =  0.  This  gives  the  relation 

<T!  (a,  —  ttj  +  tti  —  Oj)  +  <r2(a3  -  a,)  =  0. 


The  same  choice  of  the  origin  of  time  in  the  generating  solution  gives  «t  =  a2  . 
We  have  then 

<TI  (<*,  -  a,)  -f  <r2  (a,  -  a4)  =  0. 

This  equation  may  be  regarded  as  determining  o.j  in  terms  of  the  arbitrary  a,  . 
There  will  then  be  in  the  final  solution  the  two  arbitraries  c^  and  a,  besides 
the  parameter  e.  Since  at  and  e^  occur  always  in  the  combination  at  +  Oj 
they  can  be  replaced  by  a  single  arbitrary.  When  the  solutions  of  the 
periodicity  equations  are  substituted  in  (13),  the  desired  continuation  of 
of  the  first  generating  solution  is  obtained. 


OSCILLATING    >\l  KI.UTES,    EQ1  II.. \TEHAL-TRIANGLE    CASE. 

The  discussion  of  the  existence  of  the  continuation  of  the  -croud  gener- 
ating solution,  which  depend-  upon  a,  as  the  tirst  does  on  <r,  ,  differs  from 
tliat  just  given  only  in  notation,  and  will  therefore  be  omitted.  The  orbits 
arc  in  the  .ry-planc  and  have  (he  period  2r/(7,. 

159.  The  Third  Generating   Solution. — It  can  be  proved  from   the 

integral  that  the  last  equation  of  iH)  is  redundant,  and  it  will  therefore  be 
suppressed.     The  period  i<  '2*  and  the  periodicity  conditions  are 


0=0, 


(16) 


where  the  /i.  are  functions  of  n,  the  explicit  form  of  which  will  not  be 
given. 

The  first  four  equations  are  satisfied  by  a,  =  a,=  a,  =  a4  =  e  =0,  and  the 

determinant  of  the  terms  which  are  linear  in  a, , a,  is  distinct  from  zero, 

since  <rl  and  at  can  not  take  integral  values.  Therefore  these  equations  can 
be  solved  for  a, ,  a, ,  a, ,  and  a4  as  power  series  in  t,  6,  and  -y,  vanishing 
with  «.  Since  the  right  member  of  the  z-equation  in  (11)  carries  the  factor 
z.  the  last  equation  carries  the  factor  c+y.  This  factor  is  divided  out  and 
the  series  for  a,,  a,,  a,,  and  a4  are  substituted  for  these  quantities.  The 
equation  is  satisfied  by  S  =  t  =  0,  and  the  coefficient  of  5  is  not  zero.  Hence 
there  is  a  unique  solution  for  5  in  the  required  form.  This  value  of  5  is 
substituted  in  the  series  already  found  for  a,,  a,,  a,,  and  a4.  The  quantity 
•)  remains  arbitrary,  but  since  it  occurs  always  in  the  combination  0  +  7  it 
will  be  absorbed  in  the  arbitrary  constant  c.  The  periodicity  equations 
being  satisfied  in  the  required  form,  the  existence  of  the  orbits  in  question 
i-  established. 

For  an  orbit  re-entering  after  m  revolutions  the  periodicity  equations 
have  a  unique  solution  except  when  n  is  such  that  niff1  =  nil  or  ma,  =  ?», , 
where  m,  and  mt  are  integers.  It  will  be  shown  later  that  the  orbits  exist 
uniquely  in  these  cases  also.  Since  these  orbits  include  as  a  special  case  those 
re-entering  after  one  revolution,  it  follows  that  no  new  orbits  are  obtained. 

160.  The  Fourth  Generating  Solution. — In  this  case  /tt  is  restricted 
to  those  values  for  which  /«,<r,  =  TO,^,  .  Consequently  the  period  is 
2//j1T/ff1  =  2m,T/<r, .  Just  as  in  the  case  of  the  first  generating  solution,  we 
must  put  7  =  0  in  order  to  satisfy  the  last  two  periodicity  equations.  The 


310  PERIODIC    ORBITS. 

second  equation  of  (14)  can  be  suppressed  because  of  the  integral.  The 
required  terms  of  the  series  for  the  xt  are  found  from  equations  (11),  and 
the  explicit  forms  of  the  periodicity  conditions  are 


=  0, 


=  0, 


=0, 


(17) 


where  the  Bi}  are  functions  of  /i  which  will  not  be  given  explicitly. 

The  first  equation  of  (17)  is  solved  for  5  and  the  result  substituted  in 
the  other  two  equations.     After  dividing  them  by  e2,  they  have  the  form 


-]=0, 
-]=0, 


(18) 


where  the  At)  are  functions  of  /*.     In  order  that  solutions  of  the  required 
form  shall  exist,  it  is  necessary  that 

alat]»*Q.  (19) 


These  equations  are  satisfied  by  a3  =  a4  =  0.  When  these  conditions  are 
imposed,  equations  (18)  can  be  solved  uniquely  for  a3  and  a4  as  power  series 
in  e,  vanishing  with  e.  The  generating  solution  is  reduced  to  that  con- 
sidered in  §  158,  and  it  is  now  possible  to  supply  the  proof  for  the  exceptional 
cases  which  were  not  covered  by  the  previous  discussion. 

For  an  orbit  re-entering  after  one  revolution  the  proof  did  not  include 
the  case  when  o-1  =  <r2  .  When  m1  =  mt=l  the  discussion  just  given  supplies 
this  deficiency.  For  an  orbit  re-entering  after  m  revolutions,  the  case  when 
mfft  =  <rl  was  not  included.  By  putting  ml  =  m  and  mt  =  \,  we  have  the 
desired  proof. 

The  corresponding  cases  arising  from  the  second  generating  solution 
can  be  treated  by  so  combining  the  periodicity  equations  that  the  equations 
(19)  carry  the  factors  at  and  a2  respectively.  The  discussion  is  then  the 
same  as  that  just  given. 

In  order  that  equations  (19)  may  be  satisfied  by  values  of  the  at  different 
from  zero,  it  is  necessary  that  the  determinant  of  the  Atl  should  vanish. 
This  determinant  can  be  developed  as  a  power  series  in  V/I.  If  it  is  identi- 
cally zero  in  ju,  each  coefficient  of  this  series  must  vanish.  The  coefficient 


OSCILLATING    SATELLITES,    EQUILATERAL-TRIANGLE   CASE.  311 

of  V~il  was  computed  and  found  to  be  different  from  zero.  For  the  special 
values  of  M  under  consideration  here  it  may  beposMble  to  make  this  determi- 
nant vanish.  l>ut  1  it-cause  of  its  complicated  charaeter  this  possibility  has  not 
been  considered.  The  question  of  the  existence  of  these  orbits  is  thus  left 
open,  but  it  seems  improbable  that  the  necessary  conditions  can  be  satisfied. 

161.  The  Fifth  Generating  Solution.  —  The  values  of  n  in  this  case  are 
such  that  the  period  of  the  generating  solution  is  2wT  =  2wilT/<r1  .  The  last 
equation  of  (14)  is  suppressed.  The  remaining  equations  have  the  following 
form  : 


+  a,)o4-|-au(at+a,)a,-|-a14(a,-|-  ega. 

1-|-aI)(a,+oJ)1 

•••  =0, 


-|-614(a1+a,)>+&t$(al-|-a1)(c+T)1+&i,(a1+at)(c+7)i-f  •   •]  +  •••  -0, 


0, 


0, 


(20) 


where  the  at)  ,  .  .  .  ,  e,,  are  functions  of  /i  which  are  readily  determined. 
The  last  three  equations  are  solved  for  a,  ,  a4  ,  and  5,  and  the  results  thus 
obtained  are  substituted  in  the  first  and  second  equations.  After  dividing 
by  e*,  these  equations  have  the  form 


a.)  («1+a1)'+D(ai+o,)(c+7)f+«(  •••)-•« 


312  PERIODIC    ORBITS. 

In  order  that  solutions  of  (21)  of  the  required  form  shall  exist,  it  is 
necessary  that 

O.  (22) 


These  equations  are  satisfied  by  a^  =  o2  =  0.  With  Oj  and  o2  having  this  value, 
equations  (21)  then  have  a  unique  solution  for  a,  and  a2 .  But  the  generating 
solution  has  reduced  to  that  considered  in  §159.  The  orbit  obtained  is, 
therefore,  the  continuation  of  the  third  generating  solution  re-entering 
after  m  revolutions,  and  moreover  the  value  of  n  is  such  that  mal  =  m^ , 
where  m1  is  an  integer.  Thus  we  have  a  proof  of  the  existence  of  an  orbit 
in  one  of  the  exceptional  cases  omitted  in  discussing  the  third  solution. 

In  order  that  equations  (22)  may  have  a  solution  for  which  c^ ,  a2 ,  and  c 
are  different  from  zero,  it  is  necessary  that  the  determinant  of  the  A,  B, 
C,  and  D  shall  vanish.  This  determinant  can  be  developed  as  a  power  series 
in  V/I.  If  it  is  identically  zero,  each  coefficient  in  this  development  sepa- 
rately must  vanish.  The  coefficient  of  \/ju  was  computed  and  found  to  be 
different  from  zero.  For  special  values  of  M  it  may  be  possible  to  satisfy 
(22)  by  values  of  a: ,  o2 ,  and  c  which  are  distinct  from  zero.  Because  of  the 
complicated  character  of  the  coefficients,  this  possibility  has  not  been 
established.  As  in  the  preceding  case,  the  existence  of  orbits  of  this  type 
seems  improbable,  but  complete  proof  is  lacking. 

The  discussion  for  the  sixth  generating  solution  differs  only  in  notation 
from  that  just  given.  No  new  orbits  are  found,  but  a  proof  is  obtained  of 
the  existence  of  the  continuation  of  the  third  generating  solution  re-entering 
after  m  revolutions,  when  fj,  is  such  that  maz  =  m2 .  This  is  another  excep- 
tional case  not  treated  in  the  discussion  of  the  third  generating  solution. 

162.  The  Seventh  Generating  Solution. — The  values  of  /j,  for  this  case 
are  such  that  the  period  is  2w7r  =  2rn17r/<r1  =  2w27r/<r2.  As  in  the  previous 
case,  the  last  equation  of  (14)  is  suppressed.  The  remaining  periodicity 
equations  have  the  following  form: 

•J+2TO7T  [^(aj  +  aJ^Oj+tta) 

=  0, 

*21(a1  +  a1)(a2+a2)2 

=  0, 
(a,  +  a1)(at+a2)(al  +  ai; 


e2+   •   •   •    =0, 
2mvd  +  WTT  F>H  (a,  +a,)  (aj  +  a,)  +^62  (a,  +  a,)  (a4  +  a4) 

+*«(c+7)'y+  •  •  •  =o,j 

where  the  <ptj  are  functions  of  ju- 


OSCILLATING    SATELLITES,    EQUILATERAL-TRIANGLE    CASE.  313 

last  equation  of   _':;    is  solved  for  6,  and  the  result  obtained  substi- 
tuted in  the  first  four.     After  dividing  by  f,  the  equations  have  the  form 


-]=0, 


-]=0, 
-]=0, 


21 


where  the  $„  are  functions  of  p.      In  order  that  solutions  of  these  equations 
of  the  required  form  shall  exist,  it  is  necessary  that 


These  equations  are  satisfied  by  a,  =  a,  =  a,  =  a4  =  0.  With  these  values 
equations  (24)  can  be  solved  in  the  required  form  for  a,  ,  a,  ,  a,  ,  and  a4  .  The 
orbit  obtained  is  the  continuation  of  the  third  generating  solution,  and 
re-enters  after  m  revolutions.  Moreover,  n  is  such  that  ?n<r1  =  m,  and 
niffj  =  /».  ,  where  m,  ml,  and  m,  are  integers.  This  is  the  only  remaining 
exceptional  case  not  considered  in  discussing  the  third  generating  solution. 
It  has  now  been  shown  that  the  continuation  of  the  third  generating  solution 
re-entering  after  m  revolutions  exists  for  all  values  of  p. 

In  order  to  obtain  the  continuation  of  the  seventh  generating  solution, 
it  must  be  possible  to  satisfy  (25)  by  values  of  the  a,  and  c  which  are  different 
from  zero.  On  eliminating  these  quantities,  two  functions  of  the  $„  are 
obtained  which  must  vanish  if  the  non-vanishing  solutions  exist.  These 
functions  can  be  developed  as  power  series  in  %//*•  If  they  arc  identically 
zero  each  coefficient  must  separately  vanish.  The  coefficient  of  VJi  was 
computed  for  one  of  the  developments  and  found  to  be  different  from  zero. 
It  follows,  then,  that  equations  (25)  can  not  in  general  be  satisfied  in  the 
required  way.  For  special  values  of  n  this  may  be  possible,  but  on  account 
of  the  complicated  character  of  the  $„  the  possibility  has  not  been  proved. 

163.  Construction  of  the  Solutions  in  the  Plane.  —  In  constructing  the 
orbits  in  the  plane  it  has  been  found  convenient  to  use  the  normal  variables 
which  were  introduced  in  the  discussion  of  the  existence.  The  differential 
equations  are  the  first  four  of  (11),  and  the  solutions  are  given  by  (13), 
when  the  quantity  7  has  been  put  equal  to  zero.  It  has  been  shown  that 
the  quantities  a,,  a,,  a4,  and  6  can  be  determined  as  power  series  in  «, 


314  PERIODIC    ORBITS. 

vanishing  with  e,  so  that  the  corresponding  solution  shall  be  periodic, 
while  the  quantity  at  still  remains  arbitrary.  When  these  series  are  sub- 
stituted in  (13),  the  expressions  obtained  for  xly  x2,  x3,  .r4  can  be  rearranged 
as  power  series  in  e  which  converge  for  sufficiently  small  values  of  e.  We 
have  then 


I  ;  %3 %3i)    I    3-31  ^   I   «T|  '      I 


(26) 


where  the  xi}  are  functions  of  r. 

It  has  been  shown  that  the  series  (26)  have  the  following  properties: 

(a)  They  satisfy  the  differential  equations  identically  in  e. 
(ft)  Each  xt]  is  periodic  with  the  period  2ir/al  for  one  set  of  orbits,  and 
with  the  period  27r/°2  f°r  the  other  set.   | 

(c)  We  have  supposed  that  x'  =  0  at  T  =  0,  and  therefore  it  follows  that 

ffl(xl—  x1)+ffi(x!t—  £4)=0  at  r  =  0. 

(d)  The  arbitrary  aj  occurs  always  with  the  arbitrary  at  in  the  combi- 

nation at  +  at  ;  there  will  be  no  loss  of  generality  if  at  is  specialized. 
It  will  be  supposed,  then,  that  at  is  taken  so  that  at  r  =  0 


The  differential  equations  in  the  normal  variables  are 


(27) 


The  series  (26)  are  now  substituted  in  these  equations,  and  the  coefficients 
of  the  corresponding  powers  of  e  are  equated.  The  resulting  equations  are 
solved  for  the  xtj,  and  the  periodicity  conditions  are  imposed. 

The  construction  will  be  made  first  for  the  orbits  with  the  period 
The  terms  independent  of  e  are  given  by  the  equations 

Xlo      CTj  V  —  1  Xjo =  U,  £30      <T2  V       1  3-jo      U, 


OSCILLATING    SATELLITES.    EQUILATKHAL-TRIANGLE    CARE.  315 

The  general  solution  of  these  equations  is 


On  applying  condition  (fc),  it  is  found  that  aK=  a«,  =  0,  and  from  (c)  and  (ef) 
that  a10  =  0*  =  a/2.     The  solution  satisfying  the  conditions  is.  then, 


/> 
2 


which,  expressed  in  terms  of  the  original  variables  by  (10),  becomes 


The  coefficients  of  the  first  power  of  «  are  given  by  the  equations 
*i,-«r,v'=7  xu=  +^^^1  «,«„+  A,  X?+  B,  Y?, 


.  Y?, 


where  A'^01  and  Y™  represent  the  expressions  obtained  by  substituting  xa 
and  r/0  for  x  and  y  in  A',  and  Yt  .  In  order  that  the  solution  of  the  first 
equation  shall  be  periodic,  the  coefficient  of  e»">^'r  in  the  right  member 
must  vanish.  Otherwise  non-periodic  terms  of  the  type  re'^^ir  will  be 
introduced.  For  the  same  reason  the  coefficients  of  e~"v-'T,  e"v'~lr,  and 
e-<r,^=ir  m  the  right  members  of  the  second,  third,  and  fourth  equations 
respectively  must  vanish.  All  the  terms  of  this  type  come  from  the  first 
terms  of  the  right  members,  since  the  other  terms  are  of  the  second  degree 
in  TIO,  ZK,  XM,  andzM.  Since  xM  =  x4t  =  0,  these  conditions  are  satisfied  in 
the  third  and  fourth  equations.  Since  we  have  at  our  disposal  the  undeter- 
mined quantity  6,  ,  the  desired  result  is  obtained  in  the  first  and  second 
equations  by  putting  6,  =  0. 

The  equations  are  now  integrated  and  conditions  (6)  ,  (c)  ,  and  (d)  are 
imposed.  The  details  of  this  work  will  not  be  given.  The  results  expressed 
in  the  original  variables  are 


2'.. , 


n  COSO-,T+  &u  sin<r,T+6,,  cos2alT+b'M  sin2<r,r, 


316 

where 


27    (l  —  u) 


PERIODIC    ORBITS. 

-2^)  B 


-(a  +a]       a'  -    -  2a' 

un~       \"io  i  "12^  >      uii—       ^Ui2 


h  , 
bl°- 

7  /  _  SQ^OH  —  3V3(l  —  2M)«u 
°u"  4<r?  +  9 

t  L  3  V3  (1-  2/1)^11-16^1,-  (16(^ 

12(75(5(7?-1) 

i./  ,    16^  Jua+SVI  (l-2M)^i2 
12  12  a!  (So?  -1) 


.       ,          _3V3(l-2j*)a11+8or1a' 


32 


It  will  now  be  shown  that  this  method  of  obtaining  the  coefficients  of 
(26)  is  general.  Suppose  xti  and  8} >  (i=l,  .  .  .  ,  4;  j  =  0,  .  .  .  ,  n—  1)  have 
been  determined  and  that  the  xu  are  periodic.  For  the  determination  of 
xtn  and  6n  we  have  equations  of  the  following  form : 


(30) 


where  the  6tJ  and  r?0  are  known  constants.  Since  .rso  =  x40  =  0,  no  non- 
periodic  terms  can  enter  the  solutions  of  the  last  two  equations.  In  order 
that  the  solutions  of  the  first  two  equations  shall  be  periodic,  it  is  necessary 
that  the  coefficients  of  efflV-lr  and  e~fflV=[T  in  the  first  and  second  respectively 


osriu,  \TI.\I;  >\n:i.i.iiKs.   Kijrn.  \  i  KI;  \i.-i  i;i  \\<.i.i:  CASK.  317 

shall  vanish.  Tliis  gives  for  the  determination  of  6,,  the  only  undetermined 
(•mi-!.  -int.  two  ('(illations  <j,  V—  \  aSm+29n  =  0,  ff,  V—  1  a5»  —  27j,,  =0.  Since 
the  existence  proof  has  shown  that  a  is  uniquely  determined,  it  follows  that 
the.-e  equations  must  give  the  same  determination  of  5.. 

An  additional  proof  is  obtained  by  means  of  the  integral  (5').  The 
terms  of  thi^  integral  which  are  independent  of  «  are  first  expressed  in  the 
normal  variables.  Then  the  variable.-  are  replaced  hy  their  e\pn—  h>n-  a- 
power  >eries  in  t  and  the  terms  are  rearranged  so  that  the  integral  remains 
a  power  serie<  in  t.  Since  the  integral  is  an  identity  in  r  and  t,  it  follows 
that  the  coeflicient  of  each  power  of  t  must  reduce  to  a  constant  identically 
in  T.  \Ye  will  consider  the  coefficient  of  t*.  When  the  expressions  for  the 
x(,  as  functions  of  T  arc  substituted,  this  coefficient  consists  of  a  sum  of 
linearly  independent  functions  of  T.  The  coefficients  of  each  of  these 
functions  must  then  vanish. 

Let  (fl  and  <pt  denote  the  coefficients  of  e'tV=lr  and  e~"v^T  in  the  first 
and  second  equations  of  (30)  respectively.  Then,  on  integrating  these 
equations,  the  terms  in  z,,  and  x2»  carrying  ^,  and  <pt  are  found  to  be 


The  terms  in  the  coefficient  of  «"  in  the  integral  which  carry  zu  and  J*.  are 

(31) 


-2n) 

When  the  expressions  for  xw,  x^,  xtm,  and  x2.  are  substituted,  terms  carrying 
T,  re**1^^'  and  Te~*ri^/=iT  are  obtained.  All  other  terms  entering  this 
coefficient  contain  only  xtj  (i  =  1,  .  .  .  ,  4;  j  =  1,  .  .  .  ,  n—  1),  and  are 
consequently  periodic.  Hence  the  total  coefficients  of  the  above  non- 
periodic  terms  are  obtained  from  (31).  Since  the  coefficients  must  vanish, 
relations  are  obtained  which  <pl  and  <pt  must  satisfy.  The  coefficients  of 
Te*ri%=5r  anfj  Te-**,v=iT  vanisn  identically.  The  coefficient  of  T  gives  the 
relation  32  o\  (4^-1)  (<p,  +  ^)/(4<r{  +  9)  =  0.  Since  a\  >  \  ,  the  coefficient 
of  <pi-\-<f>i  does  not  vanish,  and  we  have  ^,+^  =  0.  Both  <f>i  and  <pt  carry  6. 
linearly.  Hence  if  6,  is  determined  so  that  either  of  them  vanishes,  it  follows 
that  the  other  must  vanish  also.  The  determination  of  5.  is  therefore  unique. 
Equations  (30)  are  now  integrated.  By  means  of  conditions  (6),  (c), 
and  (d),  the  new  arbitrary  constants  are  uniquely  determined  in  terms  of  the 
original  arbitrary  a  .  The  results  when  expressed  in  the  original  variables 
have  the  form 


a*> 


y»  —  2)  I b*j  cosjr-^-b'^  sinjV] , 

/-o  /-o 

«.=  - 


_2»u 2r 

<r,  V  —  1  a      <r ,  V  — 1  a 


(32) 


318 


PERIODIC   ORBITS. 


From  the  character  of  the  differential  equations  it  is  readily  shown 
that  xn  and  yn  carry  the  factor  an+1,  that  dn  carries  the  factor  a",  and  that 
a  enters  in  no  other  way.  Recalling  the  transformation  by  which  t  was 
introduced,  we  have  the  final  series 

Z  =  z0e+:r1e2+  •  •  •  ,     y  =  y0t  +  ylf?  +  •  •  •  ,     5  =  51e  +  52e2+  •  •  •  ,     (33) 

the  Xj ,  y} ,  and  dj  being  given  by  (32).  From  the  way  in  which  a  enters 
the  series,  it  is  seen  that  the  arbitraries  a  and  e  occur  always  in  the  combi- 
nation ae .  Therefore  we  can  put  a  =  1  without  loss  of  generality,  and  the 
final  series  contain  the  single  arbitrary  (. 

Y 


I-/* 


FIG.  4. 


The  construction  for  the  other  orbits  in  the  plane  differs  only  in  notation 
from  that  just  given.  The  corresponding  expressions  for  the  Xj ,  y, ,  and  5, 
can  be  obtained  simply  by  replacing  yl  by  cr2  . 

For  very  small  values  of  e  the  shape  of  the  orbits  deviates  but  slightly 
from  that  of  the  generating  ellipses.  For  ;u  =  0.01  and  e  =  0.001  two  terms 
of  (33)  were  computed  for  each  set  of  orbits. 

The  curves  found  are  given  in  Fig.  4,  the  dotted  ellipses  representing 
the  generating  solutions.  The  major  semi-axis  of  the  first  generating  solu- 
tion is  0.00111,  while  that  of  the  second  is  0.00115.  The  finite  bodies  are  at 
the  distance  unity  in  the  directions  indicated.  The  motion,  as  indicated 
by  the  arrows,  is  in  the  clockwise  direction  (the  finite  bodies  revolve  in  the 
opposite  direction) ,  the  starting-point  being  the  point  in  the  fourth  quadrant 
where  the  tangent  is  parallel  to  the  t/-axis.  The  periods  are  2 
and  27r(l  +  6)/cr,  for  the  first  and  second  solutions  respectively. 


OSCILLATING   SATELLITES,    Kt.I  1  1.  A  I  KKAL-TRIANGLK    CA-  319 

164.  Construction  of  the  Solution  with  Period  2ir.  —  The  discussion  of 
the  existence  has  shown  that  the  quantities  o,  ,  a,,  a,,  a,  ,  and  5  can  be 
determined  as  power  series  in  t,  vanishing  with  «,  so  that  x,  ,  xt  ,  xt  ,  xt,  and  z 
will  he  periodic  with  the  period  2r.  When  the  power  -<ii<s  obtained  in  this 
way  are  substituted  in  (13),  we  have,  after  re  -arraiinenient.  j-,  ,  z,,z,,  X4,  andz 
expressed  as  power  series  in  t  ,  which  converge  for  e  sufficiently  small.  By  the 
use  of  equations  (10)  ,  x  and  y  can  be  expressed  in  the  same  way.  Therefore 
the  solution  has  the  form 


The  series  (34)  have  the  following  properties: 

(a)  They  satisfy  the  differential  equations  identically  in  e. 
(6)  Each  x,,y,,  and  z,  is  periodic  with  the  period  2w. 
(c)  z(0)=0;  thereforez/0)=0  (j=0,  1,  2,  .  .  .00). 

(d)z'(0)sc;   thereforezj(0)=c,    zXO)=0          0*-l,2,  ...»). 


(35) 


The  last  property  follows  from  the  fact  that  the  arbitrary  7  occurs  always 
with  the  arbitrary  c  in  the  form  c+y,  and  can  be  put  equal  to  zero  without 
loss  of  generality. 

The  differential  equations  are 


(36) 


The  series  (34)  are  to  be  substituted  in  these  equations  and  the  coefficients 
of  the  powers  of  t  equated.     The  x, ,  y,,  and  z,  are  determined  by  solving  the 
equations  thus  obtained  and  imposing  on  the  results  the  conditions  (35). 
The  terras  independent  of  <  are  given  by  the  equations 

x:-2y'0-  f  x.-  f  v/3  (1-2/0  J/.  =  0, 

yt+2x'0-  7  v/3  (1-2/0*0  -TVo-0,  (37) 

4  £ 

ZQ   ~t~  20  =  U  . 

The  general  solution  of  these  equations  is 

(38) 


*"  v=l 


snr. 


320  PERIODIC    ORBITS. 

By  condition  (6),  we  must  have 

O01  UQJ  UQ2  O02  OQJ  Ogl  Off}  Op2  V, 

and  conditions  (c)  and  (d)  give  respectively 

C01  =  U,  C01  =  C. 

The  solution  satisfying  the  given  conditions  is,  then, 

Xo  =  0,  2/o  =  0,  z0  =  csinr.  (39) 

The  coefficients  of  the  first  power  of  e  are  given  by  the  equations 

/,=  -?  (l-2M)(l-COs2r), 

(40) 


In  order  that  the  last  equation  of  this  set  shall  have  a  periodic  solution,  it 
is  necessary  that  the  coefficient  of  sinr  shall  vanish.  Hence  we  impose  the 
condition  5i  =  0.  Upon  solving  these  equations  and  imposing  the  conditions 
(35),  we  find 


(41) 


From  the  coefficients  of  e2  we  get 


(42) 
sin3r. 

In  order  that  z2  shall  be  periodic,  the  coefficient  of  sin  r  in  the  right  member  of 
the  last  equation  must  vanish.  The  relation  obtained  determines  52  uniquely. 
The  solution  of  these  equations  satisfying  the  given  conditions  is 


(43) 


73_9(1-2M)2 


»>CILLATING    SATELLITES,    EQUILATERAL-TRIANGLE    CAM  321 

For  the  general  terms  we  proceed  by  induction.  Suppose  th&tx,,  y,,  z,, 
and  SJ(j  =  Q.  1  .....  n  —  1)  have  been  determined,  and  that  for  j  even 
it  has  been  found  that 

i 

*i  =  0,  y,  =  0,  z,=  jj  [ctcos(2*+l)T+cJsin(2*+l)T]; 

while  for./  odd,  it  has  been  found  that 


a»  cos2kr+a't  sin2&r], 


O+H/l 


*-« 

z,  =  0,  5,  =  0. 


It  can  be  readily  shown  that  when  n  is  even  the  coefficients  of  e"  are 
given  by  the  equations 


(44) 


In  order  that  the  last  equation  shall  have  a  periodic  solution  we  must  impose 
the  conditions 

-2c*.+C"=0,  CT-0. 

The  first  relation  serves  for  the  determination  of  5.  .  Since  by  the  existence 
proof  the  periodic  solution  is  known  to  exist  it  follows  that  the  expression 
Ci"  is  zero. 

An  additional  proof  that  C™  is  zero  is  obtained  by  considering  the 
integral  (5').  The  series  for  x,  y,  z  are  substituted  and  the  terms  are 
re-arranged  as  a  power  series  in  «.  Each  coefficient  of  this  series  must  reduce 
to  a  constant  identically  in  T.  Consider  the  coefficient  of  e".  The  terms 
of  this  coefficient  which  carry  zn  are 

2z'0z'm+2ztz..  (45) 

Suppose  T0  ,  .  .  .  ,  xn  ;  yt  ,  .  .  .  ,  yn  ;  z,  ,  .  .  .  ,  2..,  ;•,,...,  «._,  have 
been  determined  and  that  the  x,  ,  y,,  and  z,  are  periodic.  The  equation 
for  the  determination  of  zn  has  the  form 


COST+   •  •  •  • 

On  integrating,  the  following  non-periodic  terms  are  obtained 

zn=  —    IJT  COST  —    CJ"  r  sinr. 


322  PERIODIC   ORBITS. 

When  the  expressions  for  x} ,  y, ,  and  z,  as  functions  of  r  are  substituted  in 
the  coefficients  of  e"  in  the  integral,  the  only  non-periodic  terms  obtained 
come  from  zn .  They  are  of  the  form  T,  r  sin2r,  and  r  cos2r.  Since  the 
coefficient  of  e"  is  a  constant  identically  in  T,  it  follows  that  the  coefficients 
of  these  non-periodic  terms  are  zero.  Those  for  T  sin2r  and  r  cos2r  vanish 
identically.  The  coefficient  of  T  gives  the  relation  c  C™=Q.  Since  c^O,  it 
follows  that  <7{B)  =  0. 

Upon  integrating  (44)  and  imposing  conditions  (35),  we  find 


zn  =  cnl  cos  r+c'nl  sinr 


(46) 


The  quantities  C^+l  and  C'^  are  known  from  the  differential  equations. 
The  constants  of  integration  cBl  and  cnl  are  found  by  (c)  and  (d)  of  (35)  to 
have  the  values 

n/2  n/2 


When  n  is  odd  the  equations  obtained  from  the  coefficients  of  e"  are 

i;(n)sin2jr], 


From  the  last  equation  it  follows  at  once  that  5n  =  0,  for  otherwise  zn  will 
not  be  periodic.  Integrating  these  equations  and  imposing  the  periodicity 
equations,  we  find 


1"-  (l6j-'+9)"  -  16J-B?  +3  V3(l  -  2M)£       _  _„  , 
L  16f(4f-l)+27M(l-M) 


_  (^/2  ri6jAr+3V3(l-2M)A^-(l6j2  +  3)^    .    „  . 
L"  16/(4f-l)+27M(l-M) 


, 


2»  =0, 


(48) 


OSCILLATING    SATELLITES,    EQUILATERAL-TRIANGLE   CASE. 


323 


If  we  make  use  of  the  transformation  by  which  the  parameter  c  was 

introdiicr.l,  we  have  for  the  final  serir- 


the  .r,,  y,,  z,,  and  6,  being  given  by  (46)  and  (48).  It  is  not  difficult  to 
show  that  xn,  yn,  and  zn  carry  the  factor  c"+1,  and  Sn  the  factor  c",  and  that  c 
enters  tin  M  e\pres>ions  in  no  other  way.  Consequently  in  (49)  the  arbi- 
traries  r  and  e  occur  always  in  the  combination  c«.  Therefore  we  may  put 
c=l  without  loss  of  generality.  The  final  series  then  contain  only  the 
arbitrary  e. 

An  approximate  idea  of  the  shape  of 
the  orbit  can  be  obtained  by  considering 
the  first  two  terms  of  (49).  These  terms 
\\rre  computed  for  /u  =  0.01  and  «  =  0.5. 
The  projection  on  the  xy-p\a,ne  is  an  ellipse 
of  small  eccentricity  whose  center  is  on  the 
negative  ?/-axis  and  whose  major  axis  cuts 
the  positive  x-axis.  This  projection  is 
shown  in  Fig.  5.  The  projections  on  the 
xz  and  7/z-planes  are  shown  in  Fig.  6  and 
Fig.  7  respectively.  The  orbit  thus  con-  ~ 
sists  of  two  elongated  loops,  one  above 


FIG.  5. 


Fio.  6. 


Fio.  7. 


and  the  other  below  the  xj/-plane,  the  double  point  being  in  the  fourth 
quadrant  of  the  xy-plane. 

If  T  is  replaced  by  T+T  in  the  expressions  for  x})  y,,  and  z,,  then  x, 
and  y,  remain  unchanged  while  it  is  seen  that  z,  changes  sign.  It  follows 
that  the  loops  are  symmetrical  with  respect  to  the  xy-pl&ne,  and  that  each 
loop  is  described  in  half  the  period.  For  positive  values  of  c  the  upper  loop 
is  described  first,  and  the  motion  is  such  that  the  projection  of  the  infini- 
tesimal body  on  the  xy-pl&nc  moves  in  the  clockwise  direction.  The  period 
of  the  motion  is  given  by  the  relation  T  = 


324  PERIODIC    ORBITS. 

Orbits  about  the  Point  II. — To  each  of  the  orbits  about  point  I  there 
corresponds  an  orbit  about  point  II.  The  proofs  of  the  existence  of  these 
orbits  were  omitted,  since  they  are  similar  to  those  for  the  first  point.  The 
series  for  these  orbits  can  be  obtained  easily  from  the  corresponding  series 
for  the  first  set  of  orbits.  The  differential  equations  for  the  orbits  about 
point  II  are  obtained  from  those  for  the  orbits  about  point  I  by  changing 
the  sign  of  -\/3-  The  periodicity  conditions  to  be  imposed  are  the  same 
in  both  cases.  It  follows,  therefore,  that  the  solutions  for  the  one  case 
can  be  obtained  from  those  for  the  other  by  changing  the  sign  of  -v/3"-  There- 
fore, in  order  to  get  the  series  for  the  orbits  about  the  second  point  we  make 
this  change  in  the  series  already  obtained  for  the  first  point.  On  referring 
to  equations  (1)  and  (2),  it  is  seen  that  the  differential  equations  for  point 
I  reduce  to  those  for  point  II  if  the  signs  of  y  and  T  are  changed.  Hence 
this  transformation  can  be  made  geometrically  by  a  reflection  in  the  zz-plane 
and  a  reversal  of  the  direction  of  motion.  Thus,  it  is  easy  to  get  an  idea 
of  the  shape  of  these  orbits  from  those  already  discussed. 


CHAPTER  X. 

ISOSCELES-TRIANGLE  SOLUTIONS  OF  THE 
PROBLEM  OF  THREE  BODIES. 


BY  DANIEL  BUCHANAN. 


165.  Introduction. -  This  chapter  treats  of  periodic  solutions  of  the 
problem  of  three  bodies,  in  which  two  of  the  masses  are  finite  and  equal. 
The  thinl  I  ><  uly  is  started  at  the  initial  time  ta  from  the  center  of  gravity  of 
the  equal  masses,  and  the  initial  conditions  are  so  chosen  that  it  moves  in 
a  -traight  line  and  remains  equidistant  from  the  other  bodies.     In  I  the 
third  l)oi ly  is  assumed  to  he  infinitesimal  and  the  initial  conditions  are  so 
choM-ii  that  the  equal  bodies  move  in  a  circle  about  the  center  of  mass.* 
In  II  the  third  body  is  considered  infinitesimal  and  the  initial  conditions 
an  so  chosen  that  the  equal  bodies  move  in  ellipses  with  the  center  of  mass 
a-  the  common  focus.     In  III  the  third  body  is  considered  finite  and  the 
solutions  derived  have  the  same  period  as  those  obtained  in  I,  and  reduce 
to  those  solutions  when  the  third  body  becomes  infinitesimal. 

I.  PERIODIC  ORBITS  WHEN  THE  FINITE  BODIES  MOVE  IN  A 
CIRCLE  AND  THE  THIRD  BODY  IS  INFINITESIMAL. 

166.  The  Differential  Equation  of  Motion.— Let  wi,  and  m,  be  two 
finite  bodies  of  equal  mass,  and  n  an  infinitesimal  body.     Let  the  unit  of  mass 
he  so  chosen  that  ml  =  mt  =  1/2;  the  linear  unit  so  that  the  distance  between 
>«,  and  mt  shall  be  unity;  and  the  unit  of  time  so  that  the  Gaussian  con- 
stant shall  be  unity.     Let  the  origin  of  coordinates  be  taken  at  the  center 
of  mass,  and  the  to-plane  as  the  plane  of  motion  of  the  finite  bodie-s.     Let 
the  coordinates  of  MI,  ,  mt,  and /z  be  £,,  »;,,  0;  £,,  ij,,  0;  and  0,  0,  f  respectively. 
If  w,  and  mt  are  started  at  the  points  1/2,  0,  0  and  — 1/2,  0,  0,  respectively. 

i  hat  they  move  in  a  circle,  then 


•  \\  hen  the  finite  bodies  move  in  a  circle,  the  motion  of  the  infinitesimal  body  can  be  completely  deter- 
mined by  means  of  elliptic  integrals.  The  problem  was  first  solved  in  this  way  by  Pavanini  in  a  memoir, 
"Sopra  una  Nuova  Categoria  di  Soluzioni  Periodiche  nel  Problema  dei  Tre  Corpi,"  Annati  di  Matematica, 
Series  HI,  vol.  XIII  (1907),  pp.  179-202.  The  elliptic  integrals  obtained  by  Pavanini  were  later  developed 
independently  by  MacMillan  in  an  article,  "An  Integrable  Case  in  the  Restricted  Problem  of  Three  Bodies," 
Astronomical  Journal,  NOB.  625-626  (1911).  MacMillan  further  developed  the  solution  as  a  power  series 
in  a  parameter,  the  coefficient*  of  which  are  periodic  functions  of  I.  The  solution  obtained  in  I  baa  a 
close  relation  to  MnoMillnn'a  solution.  ais 


326  PERIODIC    ORBITS. 

The  differential  equation  for  the  motion  of  the  infinitesimal  body  is 


__          __ 

l+4f2)3/2 
where  the  accents  denote  derivatives  with  respect  to  t.     The  integral  of  (1)  is 

(2) 


where  C  is  the  constant  of  integration.  If  C  is  positive,  the  particle  M  recedes 
to  infinity.  If  C  is  negative,  the  particle  n  does  not  pass  beyond  a  finite 
distance  from  the  origin.  From  a  consideration  of  (2)  it  can  be  shown  that, 
if  C  is  negative,  the  particle  crosses  the  £ij-plane.  Hence  the  initial  time 
£0  can  be  chosen,  without  loss  of  generality,  as  the  time  when  the  particle 
is  in  the  ^-plane.  It  can  also  be  shown  from  (2)  that  if  C  is  negative, 
the  motion  of  the  particle  is  periodic,  and  that  the  period  can  be  expressed 
as  a  power  series  in  the  initial  velocity  of  n,  whose  limit  is  27r/\/8  as  the 
initial  velocity  approaches  zero.  We  shall,  however,  prove  the  existence  of 
a  periodic  solution  of  (1)  by  a  different  method. 

167.  Proof  of  Existence  of  a  Periodic  Solution  of  Equation  (1).  —  In 
this  proof  it  is  convenient  to  make  the  transformation 

,  (3) 


where  5  is  to  be  determined  so  that  the  solution  of  (1)  shall  be  periodic  in  T 
with  the  period  2w.     At  T  =  0  let 

r  =  0,  f  =  a,  (4) 

where  f  =  d^/dr.     Let  us  make  the  further  transformation 

t  =  az;  (5) 

then  when  (3)  and  (5)  are  substituted  in  equation  (1),  we  obtain 

.  •  _  (1  +  5)2  ,6) 

22 


where  z  is  the  second  derivative  of  z  with  respect  to  r.     The  initial  conditions 
for  z  become,  as  a  consequence  of  (4)  and  (5), 

2(0)  =0,  2(0)  =  1.  .  (7) 

For  \a\  sufficiently  small  the  right  member  of  (6)  can  be  expanded  into 
the  series 

z=-(l  +  5)z{l+(-i3/2)4aV+   •  •  .    +  (-3/2)(4aV)<+  •  •  •  },  (8) 
where 


/-3/2\ 

V      i      )   ' 


ISOSCELES-TRIANGLE  SOLUTIONS. 

Equation  (8)  can  be  integrated  as  a  power  series  in  a1  and  6  which  converges 
for  O^T^2r,  provided  |a|  and  |6|  are  sufficiently  small.  Let  us  write  this 
solution  in  the  form 

z=2    2  zu  5'a".  (9) 

1.0    /«• 

The  initial  values  of  the  z«,,  as  determined  from  (7)  are 

*u(0)=0,  (i,j-0,  ...  oo),  I 

J 

Upon  substituting  (9)  in  (8)  and  equating  the  coefficients  of  the  various 
powers  of  5  and  a1,  we  obtain  differential  equations  from  which  the  zu  can 
be  determined  so  that  the  initial  conditions  (10)  shall  be  satisfied. 

The  differential  equation  for  the  term  independent  of  5  and  o1  is 


and  the  solution  of  it  which  satisfies  (10)  is 

zM=sinr. 
The  differential  equation  for  the  term  in  &  alone  is 

*i*+«i*=  -sinr, 
and  the  solution  of  it  which  satisfies  (10)  is 


The  solution  of  equation  (8)  is  therefore 

z  =  sinT  +  S  [|COST  —  |sinr]  +  terms  of  higher  degree  in  a1  and  6.       (11) 

With  the  initial  conditions  (7),  the  variable  z  is  an  odd  function  of  T, 
and  therefore  a  sufficient  condition  that  it  shall  be  periodic  with  the  period 
2x  in  T  is  Z(T)  =  0.  With  the  value  of  z  given  in  (1  1),  this  condition  becomes 

0=  -  |a+termsof  higher  degree  in  a*  and  6.  (12) 

Since  the  coefficient  of  5  is  different  from  zero,  this  equation  can  be  solved 
uniquely  for  6  as  a  power  series  in  a*,  vanishing  with  a*.  Let  us  denote 
this  solution  for  5  by 

5=25,,  a1'-  (13) 

i-\ 

When  (13)  is  substituted  in  (9),  we  obtain 

z=zf,a",  (14) 


328  PERIODIC    ORBITS. 

which  converges  for  |a|  sufficiently  small.     Since  the  periodicity  condition 
has  been  satisfied,  z  is  periodic  in  r  with  the  period  2ir.     Hence 

z2J(T+2ir)=z2j(T)  (j  =  0,  ...  QO).          (15) 

The  initial  values  of  z2i  as  determined  from  (7)  are 

Z2,(0)=0  (j  =  0,  .;.«),  ] 

i    (lo) 

Z0(0)  =  l,  2,,(0)=0  (]  =  !,  ...    oo).  j 

168.  Direct  Construction  of  the  Periodic  Solution  of  Equation  (1).  —  Let 
us  substitute  (13)  and  (14)  in  (8)  and  equate  the  coefficients  of  the  various 
powers  of  a2.  Since  the  result  is  an  identity  in  a2,  there  is  obtained  a  series 
of  differential  equations  from  which  the  coefficients  of  the  solution  (14)  can 
be  determined. 

The  differential  equation  for  the  term  independent  of  a2 


2  is 


and  the  solution  of  it  which  satisfies  (15)  and  (16)  is 

z0  =  sinT. 
The  differential  equation  for  the  term  in  a2  is 

z2+z2=-52z0+6zJ=(-52+|)sinr-|sin3T. 

The  term  sinr  gives  rise  to  a  non-periodic  term  in  the  solution,  and,  in 
order  that  (15)  shall  be  satisfied,  its  coefficient  must  be  zero.     Hence 


and  the  solution  for  z2  satisfying  (16)  is 

z,=  J(sin3r- ! 
The  differential  equation  for  the  term  in  o4  is 

«4+«4=  -  (54+^)sinr+6sin3r-  |sin5r. 
In  order  that  (15)  shall  be  satisfied,  54  must  have  the  value 

5  =  —  —  > 
and  the  solution  for  z4  is  found  to  be 

z4=r^[431sinT-192sin3T+29sin5T], 

3EBQ 

where  the  constants  of  integration  have  been  determined  so  as  to  satisfy  (16). 


I8O8CEI.KS    lltl  VM.l.K    SOLUTION-  ii'J'.l 

-  far  as  computed,  it  has  been  found  that  the  St)  are  uniquely  deter- 
mined b\  -the  periodicity  and  the  initial  conditions,  and  t  hat  each  z,,  is  a  sum 
of  sines  of  odd  multiples  of  T.  the  highest  multiple  being  '2j+  1.  \\  .  -liall 
now  show  by  an  induction  to  the  general  term  that  all  the  &tj  are  uniquely 
determined  by  the  same  conditions,  and  that  all  the  ztl  have  the  properties 
which  have  been  stated.  Let  us  assume  that  St,  .  .  .  ,  $„_,;  zt,  .  .  .  ,  z,,_, 
have  been  uniquely  determined,  and  that  each  zst(k  =  0,  .  .  .  ,  j—  1)  is  a 
sum  of  sines  of  odd  multiples  of  T,  the  highest  multiple  being  2k+l.  From 
these  assumptions  and  the  differential  equations  it  will  be  shown  that 
6,,  and  z,,  are  uniquely  determined,  and  that  z,,  is  a  sum  of  sines  of  odd 
multiples  of  r,  the  highest  multiple  being  2j+l. 

Let  us  consider  the  term  in  a".    The  differential  equation  is 

,  (17) 


where  Zt;isa  known  function  of  2^(1  =  0,  .  .  .  ,j—  1)  and  5tt  (k  =  l,  .  .  .  ,j—  1). 
The  general  term  in  Ztl  has  the  form 

T   ._-*',  .A  st 

lt>~\  \6" 

where  X,  ,  .  .  .  ,  Xt  ;  ^,  ,  .  .  .  ,  M»  ',  P,  and  q  are  positive  integers  (or  zero) 
having  the  following  properties: 

(a)      /*,+  •••  +M*  is  an  odd  integer, 

(6)      MlX,+  •  •  •  +/I.X.+P.+  •  •  •  +nt-l+qp  =  2j, 

(c)      q  is  0  or  1. 

Since  each  z0  ,  .  .  .  ,  z,,_,  is  a  sum  of  sines  of  odd  multiples  of  T,  it 
follows  from  (a)  that  Ztl  is  a  sum  of  sines  of  odd  multiples  of  T.  The 
highest  multiple  is 


The  highest  value  of  NtJ  is  obtained  when  9  =  0  and  is,  therefore,  2J+1. 
Hence  (17)  has  the  form 

^+z,/  =  [-$,,+a|1>>nT+aJa/)sin3T+  •  •  •  +at%sin(2>+l)r,     (18) 
where  n{l/>,  .  .  .  ,  a£+,  are  known  constants.     From  (15)  it  follows  that 

S       -n«'n 

5,,-a,   . 
The  solution  of  (18)  satisfying  (16)  is  therefore 

z1/  =  /ll(1/)sinT+^1a/)8in3T+  •  •  •  -f  Xi 
where 


<*-• * 


330  PERIODIC    ORBITS. 

Hence  the  periodic  solution  of  (1)  in  terms  of  the  variable  T  is 


It  is  a  power  series  in  odd  powers  of  a  with  sums  of  sines  of  odd  multiples 
of  T  in  the  coefficients.  The  highest  multiple  of  T  in  the  coefficient  of  a2t+1 
is  2&-fl.  In  the  sequel  we  shall  call  such  a  series  a  triply  odd  power  series. 
The  period  in  T  is  27r,  and  in  t  it  is 


II.     SYMMETRICAL  PERIODIC  ORBITS  WHEN  THE  FINITE  BODIES 
MOVE  IN  ELLIPSES  AND  THE  THIRD  BODY  IS  INFINITESIMAL. 

169.  The  Differential  Equation.  —  Let  m^  and  w2  represent  the  two 
finite  bodies  and  ju  the  infinitesimal  body.  Let  the  system  of  coordinates  be 
chosen  as  in  §  166.  Let  the  unit  of  mass  be  so  chosen  that  ml  =  mz  =  1/2, 
and  then  let  the  linear  and  time  units  be  so  determined  that  the  mean  dis- 
tance from  Wi  to  m2  and  the  gravitational  constant  are  each  unity.  With 
these  units  the  mean  angular  motion  of  the  bodies  also  is  unity. 

Let  M  be  started  from  the  center  of  gravity  of  mt  and  m2  perpendicu- 
larly to  the  plane  of  their  motion  when  they  are  at  apsides  of  their  orbits, 
which  can  be  assumed  to  lie  on  the  £-axis.  From  the  symmetry  of  the 
motion  with  these  initial  conditions,  it  follows  that 

£2  =  ~~  £1  >  ^2  =  ~~  li  • 

Let  the  motion  of  the  finite  bodies  be  referred  to  a  system  of  axes 
rotating  about  the  f-axis  with  the  uniform  velocity  unity.  The  coordinates 
referred  to  the  rotating  axes  are  defined  by 

o;,  =  £,cos£+??«sin£,        !/«=—&  sin  <+!?«  cos  J,        z  =  r          ft  =  1,2). 

The  xt  and  yt  are  determined  by  the  conditions  that  ml  and  mt  shall  move 
in  ellipses  and  be  at  apsides  at  t  =  ta,  which  in  this  case  is  put  equal  to  zero. 
Then  it  follows,  from  the  properties  of  elliptic  motion,  that 


-t),  (20) 

where 

r  =m[l  — 


v  = 

7n  =  m,  =  m.=  ^>         e  =  eccentricity  of  ellipses. 


ISOSCELES-TRIANGLE   SOLUTIONS.  331 

The  differential  equation  for  the  motion  of  /i  is 

//         mz     mi         2m  z 
2  =- -JT--T  =  -— r-»  (21) 

M  <*  r\ 

where 


When  we  substitute  the  values  of  x,  and  yt  from  (20),  equation  (21)  becomes 

.//_  — 2mz 

(22) 


' 


Win  -re  //i  occurs  in  the  denominator  we  shall  substitute  ita  value  1/2, 
but  in  the  numerator  we  shall  make  the  substitution  m  =  m0+X,  and  con- 
sider X  as  a  variable  parameter  while  m  and  mt  both  remain  fixed.  In  order 
to  obtain  the  solution  of  the  physical  problem  we  must  put  X  =  m  —  »i0  in 
the  final  results.  With  these  substitutions,  (22)  becomes 

•"—  (mrHOSAw**1,  (23) 

1-9 

where 


and  where  each  #2/+i  is  a  power  series  in  e  with  cosines  of  integral  multiples 
of  /  in  the  coefficients,  the  highest  multiple  being  the  same  as  the  exponent 
of  the  eccentricity  e. 

170.  Determination  of  the  Period  by  a  Necessary  Condition  for  a 
Periodic  Solution  of  (23).  —  If  the  motion  is  periodic,  let  the  period  be  denoted 
by  T.  Since  the  period  of  motion  of  the  finite  bodies  is  2r,  we  must  have 

T  =  2yr,  (24) 

where  v  is  an  integer  which  denotes  the  number  of  revolutions  made  by  the 
finite  bodies  in  the  period  T. 

Let  us  take  the  initial  conditions 

2(0)  =  0,  z'(0)  =  a.  (25) 

With  these  initial  conditions  it  can  be  shown  from  (23)  that  z  is  an  odd 
function  of  t.  Hence  if  M  is  started  from  the  £i?-plane  when  m,  and  wi,  are 
at  apsides  of  their  orbits,  a  necessary  and  sufficient  condition  that  z  shall  be 
periodic  with  the  period  T  is 

z(772)=0.  (26) 


332  PERIODIC    ORBITS. 

In  order  to  determine  the  period  T  from  the  condition  (26),  we  inte- 
grate equation  (23)  as  a  power  series  in  a  and  X,  but  only  in  so  far  as  the 
term  of  the  first  degree  in  a  is  concerned.  The  differential  equation  for 
this  term  is 

i«i.o  =  0.  (27) 


This  equation  belongs  to  the  class  of  differential  equations  with  periodic 
coefficients  which  was  treated  in  Chapter  III,  where  it  was  found  that  the 
character  of  the  solutions  depends  upon  whether  or  not  4\/m^  is  an  integer. 
Since  m0  depends  upon  the  way  in  which  m  is  separated  into  ra0+X,  and  since 
|X|  must  be  taken  small  in  order  that  certain  solutions  appearing  in  the 
sequel  shall  be  convergent,  the  value  of  w0  is  in  the  vicinity  of  1/2.  We 
may  therefore  regard  4-v/m^  as  not  an  integer,  and  when  it  is  not  an  integer 
the  general  solution  of  (27)  is 


2  ,  (28) 

where  wt  and  w2  are  conjugate  complex  functions  of  the  form 

oo          n 

;fl)  (coskt  -  1)]  e", 
in)  (coskt  -  1)]  e", 


The  A(®  and  A™  are  constants  of  integration;  the  a™  and  6*'  are  real  con- 
stants which  depend  upon  the  coefficients  of  the  various  powers  of  e  in  E\  ; 
and  a  is  a  power  series  in  e  with  real  constant  coefficients,  determined  by 
the  condition  that  wt  and  M,  shall  be  periodic  with  the  period  2-rr. 
From  (25)  we  have 

A?  +A?  =  0,         [<rv^T+  u(  (0)]  ^[0)  +  [  -  «r-/=T  +u(  (0)]  A™  =  a.         (29) 


Since  i£(Q)  =  —u[(G),  the  determinant  of  the  coefficients  of  A(®  and  Af  in 
(29)  is 

A=-2[<rV^T+w;(0)]  =  v^TA1,  (30) 

where  A!  is  real,  and  it  is  different  from  zero  because  A  is  the  determinant 
of  a  fundamental  set  of  solutions  at  T  =  0.  The  solutions  of  (29)  for  Af  and 
Af  are 


Since  ^.{0)  vanishes  with  o,  and  conversely,  it  is  convenient  to  integrate 
(23)  as  a  power  series  in  A®  and  X. 


ISMM  KI.KS  THI  \M.I,E    SOLt'Tlo.V-. 

The  form  of  the  solution  of  (23)  arranged  as  a  pwvrr  >erie>  in  .I,01  is 

r>  \,e;  I), 


where  /'  i-  ;i  power  -erie-  in  .!  .'.  ,\.  and  e.     Upon  imposing  the  condition 
(26)  that  :  >!ia!l  he  |)eriodic  witli  the  period  7'.  we  obtain 


Tliis  c(iuation  is  sati>fied  l>y  .11'"  =  0,  but  this  value  of  .4,""  leads  to  the  trivial 
solution  2  =  0.  In  order,  then,  that  (31)  shall  have  a  solution  for  A™  which 
i-  ditTercnt  from  /er<>.  the  coefficient  of  -AJ0'  must  he  zero.  Xow 


»,  (—j  =  1,  or  1  -fa  power  series  in  e, 


according  as  v  in  (24)  is  even  or  odd  respectively.     Hence  u^T/ty^Q  for  \e\ 
sufficiently  small,  and  we  have 


0.  (32) 

In  order  that  this  condition  for  the  existence  of  a  periodic  solution  may  be 
satisfied,  T  must  have  the  value 


where  N  is  an  integer  which  denotes  the  number  of  oscillations  made  by 
the  infinitesimal  body  in  the  period  T.  Then  it  follows  from  (24)  that 

N  =  v<r.  (33) 

If  a  is  a  rational  fraction,  v  can  be  so  chosen  that  N  will  be  an  integer.  Inas- 
much as  o-  is  a  continuous  function  of  e,  it  can  be  made  a  rational  fraction 
by  a  proper  choice  of  e  less  than  any  value  of  \e\  which  will  insure  the 
convergence  of  the  power  series.  The  numerical  values  of  N  and  v  can  be 
obtained  when  a  has  been  determined  as  a  rational  fraction. 

171.  Existence  of  Symmetrical  Periodic  Orbits.  —  Let  us  consider  the 
terms  in  (23)  of  higher  degree  in  .4*'  and  X.  The  differential  equation 
which  defines  the  term  in  Af\  is 

^l+tn9Elzl.l  =  Zl=-\ArEl(^^'ul-e-^i^.  (34) 

The  complementary  function  of  (34)  is  the  same  as  that  of  (27),  viz., 


On  using  the  method  of  variation  of  parameters,  we  obtain 

1 


334  PERIODIC    ORBITS. 

The  determinant  of  the  coefficients  of  (a{l>)'  and  (a^)'  is  a  constant,  by 
§18,  and  is  the  same  as  (30),  viz.,  A  =  V  —  1  A^O.     Therefore 


(36) 


The  integration  of  (36)  gives  non-periodic  terms  as  well  as  periodic  terms 
having  the  period  T.  We  shall  be  concerned  only  with  the  non-periodic 
terms.  Let  the  constant  part  of  —  V  —  1  E^uJ^  be  denoted  by  Pt;  it  is  a 
power  series  in  e  with  constant  coefficients  which  are  purely  imaginary,  the 
absolute  term  of  which  is  found  to  be  2V—  i/Vrn^.  Hence 


[P,  J+periodic  terms], 

a<°  =  A?  +X  A(®  [P,  Aperiodic  terms], 

where  A™  and  A™  are  constants  of  integration  which  are  to  be  so  determined 
that  2U(0)  =  z'ltl  (0)  =  0.    Then 

zia  =  X  A?  P.tie"  V=T'  ul+e-aV=Tl  u,]  +  periodic  terms.  (37) 

It  is  necessary  to  obtain  in  addition  to  this  only  the  term  in  (A(®y. 
This  term  is  obtained  from  the  differential  equation 


r  t    —  7  -       vn  v  /»3  ^QC^ 

Jiz3,o~  A  —  ~ ^0£j3z0 .  (68) 

On  forming  the  equations  analogous  to  (36),  we  have 
«)'  =  +  — 


(39) 


The  terms  not  written  in  (39)  carry  the  exponentials  e±8/<rv/-i(  (j=l,  2)  as 
a  factor  multiplied  by  the  fourth  power  of  Mt  and  w8  considered  together. 
The  integration  of  these  terms  gives  periodic  terms  with  the  period  T. 
Let  the  constant  part  of  3\^^Im0Elulul/A1  be  denoted  by  P2 ,  a  power  series 
in  e  with  constant  coefficients  which  are  purely  imaginary,  the  absolute  term 
of  which  is  found  by  computation  to  be  36V—  iVm,,.  Then  upon  integrating 
(39),  we  obtain 

a?  =  Af  +  (Af>)i[Pt<+ periodic  terms],  1 

af  =  A?  +  (A[0>)3[P2Z+periodic  terms],  ] 

where  A®  and  Af  are  constants  of  integration  which  are  to  be  determined 
so  that  z3,0(0)  =  <0  (0)  =  0 .  Hence 

z3,0=(A?))*Ptt[e''v=r['u1+e-'rV=I'Uv]+periodic  terms.  (41) 


ISO.-'  1  .1  .1  •;>    r  It  I.  \NULE    SOLUTIONS.  335 

Now  imposing  the  condition  (26)  that  z  shall  he  periodic  with  the  period 
T,  we  obtain  from  (37)  and  (41) 


•  •  •  •    (42) 

Th.-  expremOQ  [e'^^  «l(r/2)+«-"/=TrA  u,(7y2)]  is  different  from  /on. 
for  e  =  0,  and  therefore  remains  different  from  zero  for  \e\  sufficiently  small. 
Initiation  (42)  is  satisfied  by  A™  =  0,  and  hence  the  right  side  carries  A™ 
as  a  factor.  In  order  to  tind  a  solution  of  (23)  other  than  2=0,  it  is  necessary 
to  consider  A{°VO;  therefore  the  factor  A™  can  be  divided  out  of  (42).  There 
remains  a  power  series  in  X  and  A™  and,  since  P,  and  Pt  are  different  from 
zero  for  \e\  sufficiently  small,  the  terms  of  lowest  degree  are  X  and  (A™')1. 
There  are  no  terms  in  A™e  and  e  alone,  and  the  coefficient  of  (^"O'hasa 
term  independent  of  e,  viz.,  36  V  —  1  Vm9  .  Hence,  after  A™  is  divided  out, 
equation  (42)  can  be  solved  for  A™  as  a  power  series  in  ±X',  the  coefficients 
hoi  M»;  power  series  in  integral  powers  of  e. 

Two  periodic  solutions  of  (23)  therefore  exist  having  the  period  T. 
They  have  the  form 


where  Q  is  a  power  series  in  ±X*  whose  coefficients  are  power  series  in  e. 
In  the  practical  construction  of  the  solutions  it  can  be  shown  that  2  is  a  power 
series  in  odd  powers  of  X'.  This  fact  follows  also  from  the  dynamical  nature 
of  the  problem,  since  the  motion  of  n  is  obviously  symmetrical  with  respect 
to  the  :ry-plane.  The  two  solutions  are  therefore  of  the  form 


=-s 


(43) 


where  each  Zj]+i  is  periodic  with  the  period  T. 

In  §§117-118  it  is  shown  by  a  discussion,  which  is  applicable  in  this 
problem,  that  if  v  is  even  in  (24),  the  orbits  obtained  by  taking  the  two  signs 
before  X'  are  geometrically  the  same,  but  in  the  one  the  infinitesimal  body  is 
half  a  period  ahead  of  its  position  in  the  other.  If  v  is  odd,  the  orbits  for 
-r-X'  and  —X*  are  geometrically  distinct. 

By  an  argument  similar  to  that  in  §  115,  it  can  be  shown  that  it  is  pos- 
sible to  choose  \>Q  so  that  the  solutions  (43)  will  converge  for  all  0  ^  t^  T. 


172.  Direct  Construction  of  Symmetrical  Periodic  Solutions  of  (23). — Let 
us  substitute  (43)  in  (23)  and  equate  the  coefficients  of  the  various  powers 
of  X*.  The  constants  of  integration  occurring  at  each  step  are  determined 
by  the  conditions  that  the  orbits  shall  be  symmetrical  and  periodic  with 


336  PERIODIC    ORBITS. 

the  period  T.     The  condition  to  be  imposed  in  order  that  the  orbits  shall 
be  symmetrical  is  2(0)  =0,  from  which  it  follows  that 

«w+,(0)=0  tf-0,  ...»).  (44) 

It  is  necessary  to  consider  the  terms  up  to  X5/2  before  the  induction  to  the 
general  term  can  be  made. 

The  differential  equation  for  the  term  in  X*  is 


and  the  solution  of  this  equation  is  (28).     When  (44)  is  imposed 

z,  =  A?  [e°  V=I'  u,  -  e~'  v=r'  «,], 


where  A("  is  an  undetermined  constant. 

The  differential  equation  for  the  term  in  X3/2  is 


lzt  =  Zs=-E1zl-m9EliS'l.  (45) 

When  expressed  in  terms  of  t,  the  right  side  of  (45)  has  the  form 

z3=A?e?+(Arre?\  (46) 

where  the  0fl+u(i=0,  1)  are  homogeneous  and  of  degree  2i+l  in  e+ffV=*'  and 
e-<rv=ri  The  undetermined  constant  A™  is  written  explicitly  so  far  as  it 
occurs.  In  dft+"(i=Q,  1)  the  coefficient  of  [e'^^(\h[-e-''v='lC\h  differs  from 
the  coefficient  of  [e»v=i<]A[_e-<rv=i«]/.  only  in  the  sign  of  V^l  ,  j\  and  js 
being  positive  integers  (or  zero)  such  that./1+./,  =  2i+l.  These  coefficients 
are  power  series  in  e  with  V—  1  sinjt  and  cosjt  in  the  coefficients,  k  being 
the  highest  multiple  of  t  in  the  coefficient  of  e*.  If  the  exponentials  in  0g2<+1) 
are  expressed  in  trigonometric  form,  it  is  observed  that  the  0J2'40  are  power 
series  in  e  in  which  the  coefficient  of  el  has  the  form 


y=o 


where  the  cjl +1)  are  real  constants.  Hence  the  0f +I>  are  purely  imaginary. 
In  order  that  z3  shall  be  periodic,  the  constant  parts  of  the  coefficients 
of  e+av=l1  and  e~av=lt  in  u2Z3  and  M^  respective^  must  be  zero.  From 
the  form  of  ut ,  ut ,  and  Z3  it  follows  that,  when  we  equate  to  zero  the 
constant  parts  of  these  coefficients,  we  obtain  only  the  one  equation 

-A^nPH-WO'P.l-O,  (47) 

where  Pt  and  P2  are  the  power  series  which  appear  in  (42).  Equation  (47) 
is  satisfied  by  A"'  =  0,  but  this  value  of  A("  leads  to  the  solution  2  =  0 
and  is  excluded.  The  solutions  of  (47)  for  A("  which  are  different  from 
zero  are 

(48) 


ISOSi   i  1  K«j    IKIANUI.E    SOU'TIOV-v  '.M7 

where  />,  is  a  power  series  in  *  with  constant  coeflicients  which  are  real  since 
P,  and  /'.an-  l»oth  p-.irely  imaginary  and  their  absolute  terms  have  the  same 
sign.  The  absolute  tcnn  of  /;,  is  found  by  computation  to  be 


Since  .1,"  is  purely  imaginary  the  expression  for  z,  is  real.      When  the  sign 
<>f  v7^!  is  chosen  in     Iv,  the  periodic  solution  of  (23)  which  satisfies  the 
initial  condition  2(0)  =  0  is  unique. 
The  general  solution  of  (45)  is 

z^Arf^'^+Afe-'^'^+V^iW+v?},  (49) 

whprr  .I*  and  A™  are  the  constants  of  integration.  The  particular  integrals 
<?'"  and  <pf  are  respectively  of  the  same  form  as  0{"  and  0{J)  in  (46).  From 
the  form  of  ^"  and  <p™  it  follows  that 


and  imposing  the  condition  (44)  on  (49),  we  obtain 

A?+A?  =  0. 
The  solution  (49)  therefore  becomes 


(50) 


where  .1*  remains  undetermined  at  this  step.  If  A™  is  found  to  be  purely 
imaginary  this  solution  for  2,  is  real. 

From  a  consideration  of  the  terms  in  Xw,  and  then  by  an  induction  to 
the  general  term,  we  shall  show  that  A™  and  all  the  remaining  constants  of 
integration  are  uniquely  determined,  after  the  choice  of  the  sign  in  (48)  has 
lieen  made,  by  the  conditions  that  the  orbits  shall  be  symmetrical  and  peri- 
odic with  the  period  T. 

The  differential  equation  for  the  term  in  X**  is 

2;'+nj0J?l2i  =  Zt=  -[EA+3mtBttzt]-[Ett+m.Ett].  (51) 

In  order  that  2,  shall  be  periodic  the  constant  parts  of  the  coefficients  of 
e+°'/=:it  and  e~*'/='1  in  w,Zs  and  u,Z»  respectively  must  be  zero.  From  the 
form  of  t*i  i  HI  >  and  Z,  it  follows  that,  when  we  equate  to  zero  the  constant 
parts  of  these  coefficients,  we  obtain  only  the  one  equation 

^«PJB  +  Vr^TP1»0>  (52) 

where  P™'  and  P,  are  power  series  in  e  with  real  constant  coefficients  which 
are  unique  after  the  sign  of  V—  1  in  (48)  has  been  chosen.  The  absolute 
term  of  PJ"  is  found  to  be  32,  and  therefore  the  solution  of  (52)  is 

Af-V^lp.,  (53) 

where  p,  is  a  power  series  in  e  with  real  constant  coefficients. 


338  PERIODIC    ORBITS. 

With  A®  determined  as  in  (53),  the  general  solution  of  (51)  is  periodic 
and  has  the  form 

zt  =  A?  er^'Ut+A?  e-'^'Ui+V^TW+tf+v?],  (54) 

where  A®  and  A®  are  the  constants  of  integration  and  where  the  <f>fl+n 
(i  =  0,  1,  2)  are  of  the  same  form  as  the  6fi+a  (i  =  0,  1,  2),  respectively.  It 
follows  from  the  form  of  ipf,  <pf  ,  and  <pf  that,  when  the  condition  (44)  is 
imposed,  (54)  becomes 

z,  =  A?  [e+'^'Ut-e-'^'u^  +  V^TW+v?  +<??],  (55) 

where  A®  remains  as  yet  undetermined.  This  solution  for  z5  is  real  if  A[5) 
is  purely  imaginary. 

We  shall  now  make  the  induction  to  the  general  term.  Let  us  suppose 
that  A™,  .  .  .  ,  Af~3)  have  all  been  uniquely  determined  as  power  series 
in  e  with  constant  coefficients  which  are  purely  imaginary.  Let  us  also 
suppose  that  zl  ,  .  .  .  ,  zin^  have  been  uniquely  determined  and  that  they 

are  of  the  form 

i 

(*  =  l,  •  •  •  ,  n-i),     (56) 


where  the  0S+"  are  °f  the  same  form  as  the  ^2<+1)  in  (46)  i  =  Q,  .  .  .  ,  k.  It 
will  be  shown  that  Af  ~"  is  purely  imaginary  and  is  uniquely  determined 
by  the  condition  that  z2n+1  shall  be  periodic;  also  when  the  condition 
z,B+1(0)  =  0  has  been  imposed,  that  z2n+1  has  the  form 


where  the  v'S+1'  are  of  the  same  form  as  the  <f>(£+"  in  (56). 

Let  us  consider  the  term  in  xC2a+1>/2.     The  differential  equation  for  this 
term  is 

•  •  •  -         (57) 


The  part  of  Z2n+1  not  written  explicitly  involves  zl  ,  .  .  .  ,  z2n_3  to  odd 
degrees  when  considered  together.  The  only  undetermined  constant  which 
enters  Z2n+i  is  Af  ~u,  and  it  has  the  same  coefficient  in  Z2n+l  that  Af  has  in  Z5  . 
In  order  that  z2«+i  shall  be  periodic,  the  constant  parts  of  the  coefficients 
of  e+crv=lt  and  e~av=lt  in  u2Z2n+^  and  M^an+i  respectively  must  be  zero. 
Now  since  Z2n+l  is  similar  in  form  to  Z6  ,  we  obtain  only  one  equation  when 
the  constant  parts  of  these  coefficients  are  equated  to  zero.  The  form  of 
the  equation  is 

A*-»P?+  V^lP^  =  0,  (58) 

where  P2n_i  is  a  power  series  in  e  with  real  constant  coefficients.  The 
solution  of  this  equation  for  Af"""  is 

Ar-1)  =  V^Tp2n_1,  (59) 

where  p2n_i  is  a  power  series  in  e  with  real  constant  coefficients. 


ISOSCELES-TRIANGLE    SOLUTIONS.  339 

In  general,  there  are  no  other  terms  in  M,Zj,,+,  and  w^j.+i  which  yield 
non-periodic  terms  in  z,.fl.  But  since  <r  =  N/t>,  N  and  v  being  integers, 
there  are  values  of  n  for  which  other  non-periodic  terms  than  those  already 
di-cussed  can  occur.  It  follows  from  the  properties  of  Z2.+,  that 
contain-  the  term 


[   (60) 
+v^T6,  sin*+  •  •  •  +  v/=T6.  sinfa-f-  •  •  •]•  J 

where  K  is  a  constant.     Now 

e«.+Dav=7i  =  e«>^T«[c082n<r<-|->/^T  s\n2not]. 

Consequently  these  non-periodic  terms  arise  if  k  =  2nv,  k  an  integer,  or 
if  kv  =  2nN.  This  relation  is  satisfied  if  2n  becomes  a  multiple  of  v. 
Suppose  v  is  odd.  Since  v  and  N  are  taken  relatively  prime,  the  smallest 
values  of  n  and  A-  for  which  the  non-periodic  terms  in  question  can  arise 
are  n  =  v  and  A-  =  2Ar.  If  v  is  even,  AT  is  odd;  and  the  smallest  values  of  n 
and  /,  are  //  =  v/2  and  k  =  N.  The  terms  in  which  these  non-periodic  terms 
first  arise  are  multiplied  by  Xtt"+l)/te1Ar  or  X'+'e",  according  as  v  is  odd  or  even. 
After  these  terms  first  appear  they  in  general  occur  similarly  at  all  subse- 
quent steps.  When  they  are  present,  the  equation  analogous  to  (58),  in 
so  far  as  the  terms  in  u^Z^+i  are  concerned,  is 


(61) 

where  #,._,  is  a  constant  multiplied  by  ew  or  tf  according  as  v  is  odd  or 
even.  The  terms  in  «jZ2l,+,  corresponding  to  (60)  differ  from  (60)  only  in 
the  sign  of  K  and  V—i.  Non-periodic  terms  arise  from  these  terms  in  the 
same  way  as  from  (60).  The  equation  analogous  to  (58),  in  so  far  as  the 
terms  in  w,Zta+,  are  concerned,  is  the  same  as  (61).  This  equation  can  be 
solved  uniquely  for  Af*~ti  and  the  solution  is  of  the  same  form  as  (59). 
Hence  in  all  cases  A^~"  can  be  determined  by  the  symmetrical  and  the 
periodicity  conditions. 

With  A  I*1""  determined  as  in  (59),  the  solution  of  (57)  is  periodic.    The 
general  solution  of  (57)  is 


From  the  form  of  ^2+"  it  follows  that 

*C3?fl»-P, 

and  when  the  condition  (44)  is  imposed,  z,,+,  becomes 


where  A?*+0  remains  undetermined  at  this  step.     This  solution  is  real  if 
AJ*1^"  is  purely  imaginary.     This  completes  the  induction. 


340  PEEIODIC    ORBITS. 

III.    PERIODIC  ORBITS  WHEN  THE  THREE  BODIES  ARE  FINITE. 

173.  The  Differential  Equations.  —  We  shall  now  consider  the  question 
of  the  existence  of  orbits  which  are  periodic  when  n  is  finite,  and  which 
have  the  same  period  as  those  obtained  in  I.  The  question  is  one  of  deter- 
mining initial  conditions  for  ml  ,  m2  ,  and  /x  so  that  the  motion  of  the  system 
shall  be  periodic  when  M  is  finite,  and  shall  have  the  same  period  as  when 
fj,  is  infinitesimal. 

The  origin  of  coordinates  will  be  taken  at  the  center  of  mass  of  the 
system.  The  plane  passing  through  the  center  of  mass  and  perpendicular 
to  the  initial  motion  of  n  wih1  be  taken  as  the  ^rj-plane.  Let  the  coordinates 
of  mi  ,  w,  ,  and  M  be  £,  ,  17,  ,  f  t  ;  £2  ,  ^  ,  fz  ;  and  £,  77,  f  respectively.  Let  the 
values  of  £,  17,  f,  £',  V  ',  ^  i?2,  f1;  and  f2  be  zero  at  t  =  ta.  Further,  let 


Under  these  symmetrical  initial  conditions 

&=-&,        »?i=-%,        fi=r,.  (62) 

On  making  use  of  (62)  in  the  center  of  gravity  equations,  which  are 


we  have 

0.  (63) 


Hence  M  always  remains  on  the  f-axis. 

With  the  units  chosen  as  in  §  166,  the  differential  equations  are 

£"_     -III- 
£l    ~  8   »-3 


(64) 


where  rs  =  ^+'??.     Let  us  transform  (64)  by  the  substitutions 

,  (65) 


where  5  has  the  value  determined  in  I.     Then  equations  (64)  become 

[    (66) 


(1  +  /*)(!  +  8)  f 

2372' 


ISOSCBUBS-TKIAMiLE   SOLUTIONS.  .ill 

For  n  =  0  these  equations  admit  the  solutions 


where  i/  i>  the  function  defined  in  (1(J)  and  is  periodic  in  T  with  the  period  2r. 
Now  let 


(67) 

where  p,  u,  and  u>  vanish  with  ^  =  0.     When  equations  (67)  are  substituted 
in  (66),  the  differential  equations  for  p,  u,  and  w  are  found  to  be 


- 


~  «  '  ' 


liM 


The  second  equation  of  (68)  admits  the  integral 

d 


(69) 
where  d  is  an  arbitrary  constant.     Since  u  and  p  vanish  with  n,  we  substitute 


where  do  = -v/V8(l  +  5),  and  X  is  an  undetermined  constant.     On  substi- 
tuting (69)  in  (68),  we  obtain 


" 


(70) 


174.  Proof  of  Existence  of  Periodic  Solutions  of  Equations  (70).  —  For 
M  =  0  equations  (70)  admit  the  periodic  solutions  p  =  p  =  w  =  w  =  Q.  It  will 
now  be  proved  that  if  |M|  is  not  zero,  but  sufficiently  small,  equations  (70) 
admit  solutions  expansible  as  converging  power  series  in  /*,  which  vanish 
with  jt  and  which  are  periodic  in  T  with  the  period  2*-. 

Let  us  take  the  initial  conditions 


a,,        p(0)=0,        tt(0)  =  0,        tb(0)  =  a,.  (71) 

With  these  initial  conditions  it  can  be  shown  from  the  properties  of  (70),  by 
the  usual  method,  that  p  is  even  in  T  and  that  w  is  odd  in  T.    Therefore 

P(T)=P(-T),         «>(*•)  =  U>(-T), 

and  if  the  conditions 

P(T)=U;(T)  =  O,  (72) 

are  satisfied,  p  and  w  will  be  periodic  in  T  with  the  period  2r. 


342  PERIODIC    ORBITS. 

Equations  (70)  will  now  be  integrated  as  power  series  in  ^  ,  a, ,  and  p 
in  so  far  as  the  a!  and  a2  enter  linearly  in  the  solutions.  If  the  terms  of  the 
solutions  in  which  the  Oj  and  a2  enter  linearly  are  denoted  by  pl  and  wl , 
then  the  differential  equations  defining  pl  and  wt  are 


(73) 


The  first  equation  of  (73)  is  independent  of  the  second  equation.     The 
complementary  function  of  the  first  equation  is 


where  A^  and  .B!  are  constants  of  integration.  The  function  Pt  involves  \J/ 
to  even  degrees,  and  is  therefore  a  power  series  in  a2  with  cosines  of  even 
multiples  of  ^  in  the  coefficients.  The  highest  multiple  of  T  in  the  coeffi- 
cient of  a2*  is  2k.  In  the  sequel,  such  a  power  series  is  called  a  triply  even 
power  series.  The  particular  integral  arising  from  Pt  is  a  triply  even  power 
series  unless  WsU  +  S)  is  an  even  integer.  If  Wsd  +  S)  is  an  even  integer 
the  left  side  of  the  first  equation  of  (73)  has  the  same  period  as  certain 
terms  of  the  right  side,  and  the  solution  will  therefore  contain  non-periodic 
terms.  When  V1/s(l  +  5)  is  an  even  integer,  the  period  of  the  motion  of 
m1  and  mt  is  an  even  integral  multiple  of  the  period  of  the  oscillations  of  /z. 
The  mutual  attractions  of  the  three  bodies  will  then  have  a  cumulative 
effect  and  produce  non-periodic  motion.  We  therefore  exclude  from  our 
consideration  those  values  of  a  for  which  Ws(l  +  5)  is  an  even  integer. 
With  this  restriction  upon  a,  the  solution  of  the  prequation  satisfying  the 
initial  conditions  (71)  is 

T),  (74) 


where  Ct(r)  is  a  triply  even  power  series,  and  it  contains  A  as  an  undeter- 
mined constant. 

When  (74)  is  substituted  in  the  ^-equation,  all  the  terms  of  Wl  are 
known.     With  the  left  side  simplified  and  ^  =  0,  the  equation  becomes 

«>,+  [!  +  HX'Jw^O,  (75) 

where 

6,  =  -  I-  +  9  cos2r,  04  =  ^  -  48  cos2r  +  ^  cos  4r, 

£  6Z  o 

and  where  each  02*  is  a  sum  of  cosines  of  even  multiples  of  r,  the  highest 
multiple  being  2k. 


ISOSCELES-TRIANGLE   SOLUTIONS.  343 

Equation  (75)  is  one  of  the  equations  of  t'uriulion  ,  and  the  expression  ^(r) 
or  ^-,(t-  O/WsU  +  6)  !•  ,  obtained  in  (19),  is  the  generating  solution.  Two 
jirl>itr:iry  constants,  viz.,  <„  and  a,  appear  in  its  generating  solution,  and 
according  to  j^2  and  M  the  two  fundamental  solutions  of  (75)  are  obtained 

liy  taking  the  tir.-t  partial  derivatives  of  ^  with  re-pcct    to  these  constants. 
One  solution  is  therefore 


and  it  is  periodic  in  r  with  the  period  2ir.  This  solution  contains  the 
factor  —a,  and  since  it  is  multiplied  later  by  an  undetermined  constant 
the  factor  —a  may  be  absorbed  by  the  undetermined  constant.  This  solu- 
tion can  then  be  expressed  as  (see  page  330  for  ^) 


2  ^aw  =  cosT  +     af(cos3T  —  cosr)+  •  •  •  ,  (76) 


1-0 

where  titp  =  ty/dr.     Therefore  the  ^  are  sums  of  cosines  of  odd  multiples  of 
T,  the  highest  multiple  being  2./+1.     The  initial  values  of  this  solution  are 

The  other  solution  of  (75)  is  obtained  by  differentiating  the  generating 
solution  with  respect  to  the  constant  a;  hence  this  solution  is 

(78) 

where f^)  denotes  that  the  differentiation  is  performed  only  in  so  far  as  a 
\da/ 

occurs  explicitly.    Now 

?M 

kda 


and  therefore  the  solution  (78)  is 


The  initial  values  of  this  solution  are 
u>u(0)=0,         tblf(0)  =  B 


344  PERIODIC   ORBITS. 

Since  it  is  more  convenient  for  computation  to  have  a  solution  wl2  in  which 
the  initial  values  are 

wu(0)=0,        wu(0)  =  l,  (79) 

we  take  as  the  second  solution  of  (75) 


(80) 
where  x  and  A  are  found  by  computation  to  be 

00  - 

X  =  2xsX'=sinT  +  —  a2(5sinT+sin3r)+  •  •  •  , 

^=-Wf  «•+•••• 

From  the  way  in  which  x  has  been  derived,  viz., 


it  follows  that  each  x2>  is  a  sum  of  sines  of  odd  multiples  of  T  and  that  the 
highest  multiple  is  2j+l.     Further,  since 


it  follows  from  the  character  of  \j/  that  the  coefficients  of  the  cosines  and  sines 
of  the  highest  multiples  of  r  in  <pt)  and  x2^  respectively  are  equal  numerically 
and  have  the  same  sign. 

The  solutions  (76)  and  (80)  constitute  a  fundamental  set  of  solutions, 
since  their  determinant  is  unity,  and  hence  the  general  solution  of  (75)  is 

w1  =  <V+<>[x+ATd,  (81) 

where  n"'  and  n™  are  constants  of  integration. 

When  (74)  is  substituted  in  (73),  Wl  becomes  an  odd  power  series  in  a 
with  two  types  of  terms  in  the  coefficients. 

(1)  There  are  terms  not  multiplied  by  cosV1/8(l  +  5)r  which  enter 
through  pt  and  they  form  a  triply  odd  power  series.     They  have  ^  as  a 
factor  and  will  be  denoted  by  fj.  M  1  , 

(2)  The  remaining  part  of  Wt  consists  of  terms  which  are  multiplied 
by  cos  WsU  +  S)  T.     As  we  have  already  excluded  those  values  of  a  for 
which  WsU  +  S)  is  an  even  integer,  and  as  we  subsequently  exclude  those 
values  of  a  for  which  WsU  +  S)  is  an  odd  integer,  these  terms  in  Wl  do 
not  have  the  period  2ir.     In  the  direct  construction  of  the  solutions  such 
terms  do  not  appear  in  the  right  members  of  the  ^-equations.     They  appear 
as  the  complementary  functions  of  the  p,-equations  and,  since  they  do  not 
have  the  period  2tr,  they  are  excluded  by  assigning  zero  values  to  the  con- 
stants of  integration.     We  can,  therefore,  disregard  the  terms  in  W^  which 
do  not  have  the  period  2ir  and  consider  Wl  to  have  the  form 


ISOSCELES-TRIANGLE  SOLUTIONS.  345 

By  varying  the  paraim-trr-  nf  and  7^'  in  (81),  we  obtain 

(82) 


The  determinant  of  the  coefficients  of  nj"  and  nj"  in  (82)  is  a  constant 
[§  IS],  and  from  (77)  and  (79)  it  is  seen  that  the  value  is  unity.  Equation.- 
(82)  can  therefore  be  solved  for  n^'  and  n,<0,  and  the  solutions  are 


(83) 
integrating  these  equations,  we  obtain 

(84) 


The  7?"'  and  ij"'  are  the  constants  of  integration.  The  A/J"  is  a  power 
series  in  if  with  constant  coefficients.  The  A/{°  is  a  power  series  in  odd 
powers  of  a  with  sines  of  even  multiples  of  r  in  the  coefficients,  the  highest 
multiple  of  T  in  the  coefficient  of  au+I  being  2k+2.  The  Rl  has  the  same 
form  as  the  M,u>  except  that  it  has  cosines  instead  of  sines.  Since  the 
coefficients  of  awcos  (2j  +  1  )  T  and  a?sm(2j+l)T  occurring  in  <f>  and  \ 
respectively  are  equal,  so  also  the  coefficients  of  aw+lsin  (2j  +  2)  T  and 
av+1cos(2j  +  2)T  in  A/,U)  and  Rl  respectively  are  equal.  When  (84)  is 
substituted  in  (81)  the  solution  of  the  second  equation  in  (73)  is  found  to  be 

u>1  =  ,?V+i»nx+4Td-M<S,-aMrTd,  (85) 

where  Si  is  a  power  series  in  odd  powers  of  a  with  sines  of  odd  multiples  of 
T  in  the  coefficients.  From  the  form  of  the  <p,  x,  3/,0),  and  Rl  it  follows  that 
Si  is  a  triply  odd  power  series.  The  expressions  *S,  and  M*'  carry  X  as  an 
undetermined  constant.  When  the  constants  of  integration  are  chosen  so 
that  (71)  shall  be  satisfied,  the  solution  (85)  becomes 

w.-k-n  &(o)-«jfrnu-MH+*&-«*r^      (86) 

Now  let  us  impose  the  periodicity  conditions  (72)  on  the  solutions  (74) 
and  (86).     In  so  far  as  the  linear  terms  in  a,  and  a,  are  concerned,  we  get 


+  terms  in  /za,  and  higher  degree  terms  in  o,,  /uo,,  and 


t0(ir)=0=  —0,^1  TT 

+terms  in  o,  ,  M  and  higher  degree  terms  in  n,  a,  ,  and  o,. 


(87) 


These  equations  are  satisfied  by  a,  =  o,=/i  =  0,  and  X  arbitrary.    The  deter- 
minant of  the  coefficients  of  the  linear  terms  in  a,  and  a,  is 


346  PERIODIC    ORBITS. 

We  now  exclude  those  values  of  a  for  which  WsU+S)  is  an  odd  integer, 
and  as  we  have  already  excluded  those  values  of  a  for  which  V1/s(l  +  5) 


is  an  even  integer,  sinVVsd  +  d)  ir  is  not  zero  and  D  can  vanish  only 
when  a  is  zero.  In  order  to  obtain  solutions  which  are  not  identically  zero, 
a  must  be  distinct  from  zero.  Hence  the  determinant  D  is  distinct  from 
zero  and,  by  the  theory  of  implicit  functions,  equations  (87)  can  be  solved 
for  ax  and  a2  as  power  series  in  ju,  vanishing  with  fj..  The  coefficients  of  the 
various  powers  of  n  are  power  series  in  a  and  contain  additional  terms  in 
I/a*.  The  X  enters  the  coefficients  of  the  solutions  as  an  arbitrary,  but  the 
solutions  are  unique  if  A  is  assigned.  Hence  periodic  solutions  of  (70)  exist 
and  are  of  the  form 

p=  S  pi//,  w=  SW,M,  (88) 


4=1 


where  each  pt  and  wt  is  periodic  in  T  with  the  period  2ir. 

175.  Proof  that  all  the  Periodic  Orbits  are    Symmetrical.  —  Let   us 

suppose  that  the  condition  is  no  longer  imposed  that  the  equal  bodies  shall 
be  at  apsides  of  their  orbits  when  the  third  body  crosses  the  £rj-plane,  and 
let  us  consider  the  question  of  the  existence  of  periodic  solutions  of  (70), 
with  the  period  2w  in  T,  when  the  initial  conditions  are 


0l,        p(0)  =  o,,        w(0)=0,        tb(0)  =  o,.  (89) 

Sufficient  conditions  that  p  and  w  shall  be  periodic  with  the  period  2ir  are 
p(2ir)-p(0)=0,     p(2ir)-p(0)=0,     w(2w)-w(Q)=Q,     M>(2*r)-M>(0)  =  0.   (90) 

These  four  conditions  can  not  be  satisfied  by  the  three  constants  a,  unless 
one  condition  is  a  consequence  of  the  other  three.     We  now  show  that  the 
last  condition  can  be  suppressed  when  the  first  three  have  been  imposed. 
The  original  differential  equations  (64)  admit  the  integral 


|i  +const. 

When  the  substitutions  (65)  and  (67)  are  made  and  u  is  eliminated  by  means 
of  (69),  this  integral  takes  the  form 


UofTp)  +  [(i+p)2+4(i2+MW+™)2]'  +  const-J  ' 


ISOSCELES-TRIANGLE   SOLUTIONS.  347 

Let  us  make  in  (91)  the  usual  substitutions 

/>  =  p(0)+p,        />  =  p(0)+p        w  =  0+w,        w  =  w(Q)+w,        (92) 

where  /'),  p,  w,  and  »•  vanish  at  r  =  0,  and  let  us  denote  the  resulting  equation 
by  (91a).  By  putting  r  =  0,  we  obtain  from  (91a)  an  equation  (916)  con- 
nect iim  the  terms  in  (91a)  independent  of  p,  p,  w,  and  w.  When  this  equa- 
tion (916)  is  substituted  in  (91a)  there  results  an  equation  of  the  form 

F(p,p,w,w)  =  0,  (93) 

in  which  there  are  no  terms  independent  of  the  arguments  indicated.  The 
linear  term  in  w  enters  (93)  with  the  coefficient 

8/i(l  +/*)[*+»(<))], 

which,  we  shall  show,  is  different  from  zero  at  r  =  2f.  Since  the  third  body 
is  assumed  to  be  finite,  8^(1  +/*)  is  distinct  from  zero.  The  coefficient  of  w  is 
therefore  different  from  zero  at  r=  2ir  unless  u?(0)  =  —  ^(2ir)  =  —  a.  Now 
the  third  body,  when  assumed  to  be  infinitesimal,  has  the  initial  speed 
),  and  ti;(0)/VV8(l+8)  is  the  additional  initial  speed  to  be  so 


determined  that  the  orbits  shall  be  periodic  in  r  with  the  period  2ir  when  the 
t  hinl  body  becomes  finite.    If  this  additional  initial  speed  is  —  a 


then  the  whole  initial  speed  is  zero,  and  the  third  body  remains  at  the  center 
of  gravity  since  there  is  then  no  force  component  normal  to  the  £i;-plane.  In 
order  therefore  to  obtain  solutions  in  which  f  is  not  identically  zero,  we  must 
consider  tb(0)  ^  —a.  Hence  the  coefficient  of  the  linear  term  w  in  (93)  is  dis- 
tinct from  zero  at  T  =  2r,  and  therefore  (93)  can  be  solved  for  «J(2r)  as  a 
power  series  in  p(2ir),  p(2ir),  and  w(2*)  which  vanishes  with  p(2r),  p(2ir), 
and  uJ(2r)  .  This  power  series  is  unique  if  X,  which  appears  in  the  coefficients, 
is  assigned.  Hence  if  the  first  three  conditions  of  (90)  are  imposed,  then 
p(2ir)=p(2ir)=uJ(2T)=0;  and  since  w(2*)  vanishes  with  p(2r),  p(2ir),  and 
w(2r),  it  follows  that  w(2v)  =0  or  w(2ir)  -ti>(0)  =0.  Therefore  the  fourth 
equation  of  (90)  is  a  consequence  of  the  first  three  and  may  be  suppressed. 
Equations  (70)  will  be  integrated  as  power  series  in  n  and  o,(t  =  l,  2,  3), 
but  only  in  so  far  as  the  a,  enter  linearly  in  the  solutions.  The  differential 
equations  from  which  these  linear  terms  in  o<  are  obtained  are  the  same  as 
(73).  Their  solutions  satisfying  (89),  in  so  far  as  the  linear  terms  are  con- 
cerned, are 


(94) 


348  PERIODIC    ORBITS. 

When  the  first  three  conditions  of  (90)  are  imposed  on  p  and  w,  we  have  as  a 
consequence  of  (94) 


=  0,  [cos  Vi/8(l  +  5)  27T  -  1  ]  +  ^1/81  +  g)  sin  VVsI  1  +  3)  27T 

+  terms  in  ju,  jua3,  and  higher  degree  terms, 


(95) 


+ terms  in  ju,  /ia3 ,  and  higher  degree  terms, 

Q  =  2a3Air+ terms  in  o0  a.,,  /z,  and  higher  degree  terms. 
The  determinant  of  the  coefficients  of  the  linear  terms  in  at  in  (95)  is 

2r], 


This  determinant  does  not  vanish  when  a  is  not  zero,  and  consequently 
(95)  can  be  solved  for  a,  as  power  series  in  /*,  vanishing  with  /*.  These 
solutions  are  therefore  unique  if  X,  which  enters  the  coefficients,  is  assigned. 
Hence  the  periodic  solutions  of  (70)  for  p  and  w,  with  the  initial  conditions 
(89),  are  of  the  same  form  as  those  obtained  in  (88)  for  the  symmetrical 
orbits.  The  unrestricted  and  the  symmetrical  orbits  are  unique  for  a  not 
zero  and  for  any  value  of  X,  and  therefore,  since  the  unrestricted  orbits 
include  the  symmetrical  orbits,  all  the  periodic  orbits  are  symmetrical. 

176.  Direct  Construction  of  the  Periodic  Solutions  of  (70).  —  In  order 
to  construct  the  periodic  solutions  of  (70),  we  substitute  (88)  in  (70)  and 
equate  the  coefficients  of  the  various  powers  of  /*.  The  arbitrary  constants 
of  integration  are  to  be  so  determined  that  w(0)  =0  and  that  each  pf  and  wt 
shall  be  periodic  in  T  with  the  period  2w. 

The  differential  equations  for  the  terms  in  p  are 


(96) 


The  general  solution  of  the  prequation  is 


where  C^T)  is  the  same  triply  even  power  series  as  in  (74).  Since  the 
complementary  function  does  not  have  the  period  2w  when  VVs(l  +  5)  is 
not  an  integer,  the  arbitrary  constants  A^  and  Bl  must  be  zero  in  order 
that  pl  shall  be  periodic  with  the  period  2ir  .  Hence  the  desired  solution  of 
the  ^.-equation  is 

Pi-C^r).  (97) 


BWMI  KI.K-    IKIANGLE   SOLUTIONS. 


When  (<)7)  is  substituted  in  ]\\  all  the  terms  of  It",  arc  known.  The 
general  solution  of  the  (^-equation  is  the  same  as  (85)  except  that  the 
particular  integral  does  not  carry  the  factor  M.  Hence 


(98) 

Since  all  the  periodic  orbits  are  syimnetrical  we  impose  the  condition 
0    »0,  from  which  it  follows  that 

wt(Q)  =  0  (t-i,  .  .  .  oo).  (99) 

A-  a  consequence  of  (99),  the  constant  >?,"  is  zero.  In  order  that  to,  shall 
l>e  periodic,  the  right  member  of  (98)  must  contain  no  terms  in  r  except 
those  in  which  it  occurs  under  the  trigonometric  symbols.  The  non-periodic 
terms  in  1  98)  disappear  if  the  constant  t;Jl)  is  so  determined  that  A-tf  =  aM(m, 
from  which  it  follows  that 


where  Pu>(al)  is  a  power  series  in  a*  with  constant  coefficients.     When  these 
values  of  i?,<u  and  17™  are  substituted  in  (98),  the  solution  for  u;,  becomes 

*,  (ioo) 


/-o 


where  each  S™+n  is  a  sum  of  sines  of  odd  multiples  of  r,  the  highest  mul- 
tiple being  2j+l. 

The  differential  equations  for  the  terms  in  /u*  are 

(a)    P,+    (1  +  «)P,=  P,,        (6)   ^l+iX'w^WV     (101) 


The  first  equation  is  independent  of  the  second,  and  the  terms  in  P,  are 
completely  known.  The  complementary  function  of  (lOla)  does  not  have 
the  proper  period  and  is  excluded  from  the  solution  by  taking  zero  values 
for  the  constants  of  integration.  Since  Pt  has  the  period  2ir  the  particular 
integral  has  the  same  period  and  is  the  solution  desired.  The  function  P, 
is  a  triply  even  power  series  multiplied  by  I/a1,  and  since  the  particular 
integral  has  the  same  form  as  Pt  it  is  denoted  by 


/-I 


where  each  C^  is  a  sum  of  cosines  of  even  multiples  of  r,  the  highest 
multiple  being  2j. 

When  p,  has  been  obtained,  all  the  terms  in  Wt  are  known.  That  part 
of  Wt  which  is  independent  of  p,  and  wl  is  a  triply  odd  power  series.  The 
term  p,  is  multiplied  by  a  triply  odd  power  series  and  yields  a  triply  odd 
power  series  multiplied  by  I/a1.  The  terms  tc,  and  w(  are  multiplied  by  triply 
even  and  triply  odd  power  series  respectively  and  together  yield  a  triply  odd 


350  PERIODIC    ORBITS. 

power  series  multiplied  by  I/a4.     The  lowest  power  to  which  a  enters  a4  W2 
is  found  from  the  term  w\t  to  be  3.     Hence  the  form  of  W2  is 


a  ,=1 

where  each  W%i+1)  is  a  sum  of  sines  of  odd  multiples  of  r,  the  highest  multiple 
being  2./+1. 

The  complementary  function  of  (1016)  is  the  same  as  that  of  (75),  viz., 

The  general  solution  of  (1016)  has  the  same  form  as  (85)  and  is  denoted  by 

TV,  (104) 


a 

where  S2  has  the  same  form  as  W2  and  where  Mf  is  a  power  series  in  a2  with 
constant  coefficients.  The  17?'  and  rjj2'  are  the  constants  of  integration. 
The  constant  rjf  must  be  zero  to  satisfy  (99),  and  in  order  that  wt  shall 
be  periodic  tff  must  have  the  value 

1      l/f(0) 


where  P{2)(a2)  is  a  power  series  in  a2  with  constant  coefficients.     With  these 
values  of  yf  and  iff,  the  solution  (104)  becomes 


w«=—. 


where  the  S?  +1)  have  the  same  form  as  the  S?'+1>  in  (100). 

The  form  of  the  w,  is  apparent  from  (100)  and  (106).  The  form  of  the 
P>0>2)  is  not  apparent  from  (97)  and  (102),  and,  before  the  induction  can 
be  made,  it  is  necessary  to  consider  the  term  in  /x3  in  so  far  as  p3  is  concerned. 
The  differential  equation  for  p3  is 

P3,  (107) 


where  all  the  terms  of  P3  are  known.  That  part  of  P3  independent  of  the 
Wj  is  a  triply  even  power  series  multiplied  by  I/a2.  The  w}  enter  P3  multi- 
plied by  power  series  which  are  triply  odd  or  triply  even  according  as  the 
W],  considered  together,  enter  to  odd  or  even  degrees  respectively.  These 
terms  form  a  triply  even  power  series  multiplied  by  I/a4.  Since  P3  contains 
the  term  wt\l/,  the  lowest  power  of  a2  in  a4P3  is  unity.  The  complementary 
function  of  (107)  does  not  have  the  period  2ir,  and  the  solution  desired  is 
the  particular  integral.  This  solution  has  the  same  form  as  P3  and  is 
denoted  by 


where  the  Cfn  have  the  same  form  as  the  C™  in  (102). 


ISOSCELES-TRIANGLE    SOLUTIONS.  ii.'i  1 

Let  us  suppose  that  the  pt,  wt,  ijj°(t-l,  .  .  .  ,  n-1;  j-1,  2),  have  been 
computed  and  that 


(109) 


. 

/-I  /-O 


,r'   " 


where  the  C™,  SJM+I)I  and  P<V)  have  the  same  form  as  C™,  8?+°,  and 
r)  respectively.     From  these  assumptions  and  the  differential  equations 
arising  from  the  coefficients  of  /*"  it  will  be  shown  that  equations  (109)  hold 
when  i  =n.     The  differential  equations  for  the  terms  in  n*  are 

(a)     P.+(1  +  *)P.  =  P.,         (6)     w.+    l+Za".-^.         (110) 


As  in  the  previous  steps,  the  first  equation  is  independent  of  the  second. 
Since  the  right  member  of  the  first  equation  in  (70)  carries  M  as  a  factor, 
the  PJ  and  10,  which  enter  Pn  have  j<n.  Hence  all  the  terms  of  P.  are  com- 
pletely known.  The  general  term  of  P.  has  the  form 


\ 


multiplied  by  a  triply  odd  or  by  a  triply  even  power  series  according  as 
/*[+•••  +M>  is  odd  or  even  respectively.  The  X,,  X[,  .  .  .  ,  n,,  n't  are 
positive  integers  (or  zero)  such  that 

X.XI+  •  •  •  +XX+^M;+  •  •  •  +MX^n-l.  (112) 

From  the  form  of  the  general  term  it  follows  that  Pn  is  a  triply  even  power 
-•lies  multiplied  by  I/a',  where 

i  =  (2X,-2)x;+  •  •  •  +(2X.-2)x;+2(MlM;+  •  •  •  +*$.        (113) 

This  expression  is  even  and  has  its  highest  value  when  X|  =  •  •  •  =  X^  =  0,  i.  e., 
in  the  terms  of  P.  in  which  only  the  w,  appear.  Hence,  from  (112),  the 
highest  value  of  i  is  2n  —  2.  Since  P.  contains  the  term  u>,_,^,  the  lowest 
power  of  o'  in  aw~*P.  is  found  to  be  unity.  Therefore  the  form  of  P.  is 


where  the  P™  have  the  same  form  as  the  CJ".     The  only  solution  of 
(llOa)  which  has  the  period  IT  is  the  particular  integral,  and  it  has  the  form 


where  the  C™  are  of  the  same  form  as  the 


352  PERIODIC    ORBITS. 

When  pn  has  been  determined,  all  the  terms  in  Wn  are  known  since 
they  arise  from  p^  (j  <!  n)  and  wk  (k<ri).  The  general  term  of  Wn  has  the 
same  form  as  (111),  but  it  is  multiplied  by  a  triply  odd  or  triply  even 
power  series  according  as  M;+  •  •  •  +/^  is  even  or  odd  respectively.  The 
Xj  ,  AI,  .  .  .  ,  p.,  )  MJ  are  positive  integers  (or  zero)  such  that 

\\{+  •  •  •  +M;M^n.  (114) 

From  the  form  of  the  general  term  it  follows  that  Wn  is  a  triply  odd  power 
series  multiplied  by  (I/a)1,  where 


This  expression  is  even  and  has  its  highest  value,  2n,  in  the  terms  of  Wn  in 
which  only  the  w,  appear.  The  lowest  power  to  which  a  enters  a?nWn  is 
obtained  from  the  term  in  which  \l/  enters  to  the  lowest  power.  This  term 
is  w^-it,  and  therefore  the  lowest  power  of  a  in  a"nWn  is  found  to  be  3. 
Hence  the  form  of  Wn  is 


where  the  W?  +I)  have  the  same  form  as  the  S?+1\ 

The  complementary  function  of  the  second  equation  of  (110)  is 


and  by  varying  the  parameters  n{B>  and  n™  we  have  as  the  complete  solution 

-±=iM?T<p,  (115) 


where  Sn  has  the  same  form  as  Wn,  and  where  M™  is  a  power  series  in  a2  with 
constant  coefficients.  The  i}™  and  i$*  are  the  constants  of  integration. 
The  constant  if?  must  be  zero  to  satisfy  (99),  and  in  order  that  wn  shall  be 
periodic,  r)™  must  have  the  value 


(n)  _ 

1*   -- 


where  P(lt)(a2)  is  a  power  series  in  a2  with  constant  coefficients.     With  these 
values  of  the  constants  of  integration,  the  solution  (115)  becomes 


ivn  = 


where  the  <Siy+1)  have  the  same  form  as  the  Sf+l>.     This  completes  the 
induction. 


iMiSCELES-TKIANciLE   SOLUTIONS. 

177.  The  Periodic  Solution  of  Equation  (69).—  When  the  solution  for 

p  is  substituted  in  (li'.)i,  u  can  be  obtained  by  a  single  integration.  Since  p 
is  a  power  series  iii  n  with  coeflicients  which  are  triply  even  power  series 
multiplied  by  l/«r',  the  right  side  of  (69)  will  contain  terms  independent  of  r. 
After  the  integration,  u  will  therefore  contain  a  term  in  T  with  X  appearing 
in  the  coeliicient.  \Ye  shall  now  show  that  X  can  be  so  determined  that 
this  coefficient  shall  be  zero.  When  X  has  been  so  determined  u  will  be 
periodic  in  T  with  the  period  2r. 

The  solution  for  p,  obtained  in  (97),  in  so  far  as  the  term  in  X  is  con- 
cerned, is 


\\  hen  (116)  is  substituted  in  (69)  and  d  is  replaced  by  d,,+X/x,  the  constant 
terms  appearing  on  the  right  side  of  (69)  are 

—  3  Xju+  higher  degree  terms  in  \n  and  n.  (117) 

Since  X  carries  the  factor  //,  we  may  replace  \n  by  <r,  and  then  the  expres- 
sion (117)  becomes 

—  3<r+  higher  degree  terms  in  <r  and  M-  (118) 

If  (118)  is  equated  to  zero,  the  resulting  equation  can  be  solved  uniquely  for 
<r  as  a  power  series  in  n,  vanishing  with  n  (and  converging  for  \n\  and  |a| 
sufficiently  small).  Let  this  series  be  denoted  by 


from  which  the  value  of  X  is  found  to  be 

X-  S  XX  (119) 

j-t 

the  y,  and  X,  being  constants.  With  this  value  of  X  the  u  will  be  periodic, 
and  since  X  and  p  are  power  series  in  n,  the  u  is  also  a  power  series  in  n  and 
is  denoted  by 

u=  £  u,n',  (120) 

/-i 

where  each  u,  is  separately  periodic  for  |^|  sufficiently  small.  The  solution 
(120)  converges  for  |a|  and  |/u|  sufficiently  small. 

In  order  to  construct  the  u,  directly,  we  substitute  in  (69)  the  series 

1  1!>)  and  (120),  and  the  solution  already  obtained  for  p,  in  which  X  is  to  be 

replaced  by  (119).     The  various  X,  are  determined  so  that  the  right  side  of 


354  PERIODIC   ORBITS. 

(69)  shall  contain  no  constant  terms  in  the  coefficients  of  the  n1  •     The  con- 
stant term  appearing  in  the  coefficient  of  n  is 

cx> 

o  r\         "v  \  fly)  j-i^l 

—  o  LAO*~~  £t  AO    d  \9 

the  Xo2^  being  known  constants.     This  term  is  zero  if 

OO 

X0=  £  X0    a  . 

It  is  necessary  to  consider  the  terms  up  to  fj?  before  the  induction  to  the 
general  term  can  be  made.  The  constant  term  appearing  in  the  coefficient 
of  M2  has  the  form 

00 

o  r\         v*  \  (2/>  rt2.n 

—  O[A1 —   2j  At     €1   \, 

where  the  Xf^  are  known  constants.     This  term  vanishes  if  \  has  the  value 

OO 

/=0 

The  constant  term  in  the  coefficient  of  ^'  has  a  similar  form;  that  is,  it 
can  be  written 


J-O 


the  Xj2J)  being  known  constants,  and  this  term  vanishes  if  \  is  so  determined 
that 


-9  2j  ^»  a  • 


Suppose  Xo,  \  ,  .  .  .  ,  XB_i  have  been  uniquely  determined  in  the  same 
way  and  that 


J-0 


the  \™  being  known  constants.    From  the  form  of  pt  in  (115)  it  follows 
that  the  constant  term  in  the  coefficient  of  M"+I  has  the  form 


a  ], 


J=0 


where  the  X®"  are  constants  derived  from  the  p}  and  X,(t  =  l,  .  .  .  ,  n-1),  and 
are  therefore  known.    This  constant  term  is  zero  if 

^--JM 

The  same  process  of  determining  the  X,  obviously  can  be  indefinitely  con- 
tinued. 


I-"  IUIAM.1.K    SOLUTIONS. 


\\  ith  X  thus  determined  as  a  power  scries  in  p,  the  integration  of  (69) 
yields  periodic  terms  only,  and  from  the  form  of  the  p,  it  follows  that  the  ut 
have  the  form 


the  f/™  are  sums  of  sines  of  even  multiples  of  r,  the  highest  multiple 
being  2j.     The  periodic  solution  of  (69)  is'therefore 


where  t/  is  the  constant  of  integration.  Since  the  mass  m,  is  started  from 
the  point  0,  0, 1/2,  at  £  =  <„  or  at  r  =  0,  it  follows  thatt>  =  0  at  T  =  0  and  there- 
fore, from  (67),  the  value  of  u  at  r  =  0  is  zero.  Hence  the  constant  U  is  zero. 

178.  The  Character  of  the  Periodic  Solutions. — When  the  periodic  solu- 
t  ions  for  p,  u,  and  w  have  been  determined,  the  solutions  for  £, ,  17, ,  f,  (t  =  1 ,  2) 
and  f  are  obtained  by  means  of  the  equations  (67),  (65),  (63),  and  (62). 
These  solutions  are  all  periodic  in  t  with  the  period  P  =  2*-\/1/8(l-r-5)- 
Three  arbitrary  constants  appear  in  the  solutions,  viz.,  a,  /*,  and  ^. 
The  expression  a/vVsU+di)  represents  the  initial  speed  of  the  third 
body  in  I,  n  the  mass  of  the  third  body,  and  ^  the  epoch.  The  mass  n  is 
restricted  in  magnitude  but  can  be  increased  step  by  step  by  making  the 
analytic  continuation  of  the  solutions  already  obtained,  provided  the  series 
do  not  pass  through  any  singularities  in  the  intervals.  This  can  be  done 
by  the  process  already  developed.  The  parameter  a  is  restricted  in  mag- 
nitude and  so  that  -y/VsU+S)  is  not  an  integer.  As  already  stated  in 
§  166,  it  can  be  shown  from  equation  (2)  that  the  motion  of  the  infinitesimal 
body  will  be  periodic  if  the  initial  conditions  are  chosen  so  that  the  constant 
C  is  negative.  With  the  initial  conditions  chosen  as  in  (4),  the  constant  C 
has  the  value  4  j2ay(l  +  S)-l}.  Now  if  2ay(l  +  5)  =  l  or  >1,  the  infini- 
tesimal body  recedes  to  infinity  with  a  velocity  which  is  zero  or  greater 
than  zero  respectively,  and  therefore  the  motion  will  not  be  periodic. 
Hence  a  must  be  restricted  so  that  2a*/(l  +  6)<l.  The  epoch  ta  is  arbi- 
trary and  may  be  chosen  to  be  zero  without  loss  of  geometric  generality. 
Hence  for  given  values  of  |M|  and  of  |a|  sufficiently  small  and  such  that 
\A/8(1-M)  is  not  an  integer,  there  exists  one  and  only  one  set  of  periodic 
orbits  which  are  geometrically  distinct  with  the  period  P  in  I,  and  which 
reduce  to  those  obtained  in  I  for  n  =  0. 

In  proving  the  existence  of  the  periodic  solutions  of  (70),  if  the  period 
were  chosen  to  be  2vir  in  T,  v  an  integer,  it  could  be  shown  by  the  same 
process  that  periodic  solutions  would  exist  under  the  same  restrictions  on 


356  PERIODIC    ORBITS. 

a  and  /x.  The  constant  X  could  be  determined  so  that  the  solution  for  u 
would  have  the  period  2vw,  and  hence  the  periodic  solutions  of  (64)  would  be 
unique  as  soon  as  v  were  chosen.  Therefore,  for  given  values  of  a  and  of  p 
sufficiently  small,  and  such  that  Ws(l  +  5)  is  not  an  integer,  there  exists 
one  and  but  one  set  of  orbits  which  re-enter  after  v  synodic  revolutions. 
Hence  the  result  is  unique  for  every  v,  and  since  the  orbits  reentering  after 
v  revolutions  include  those  reentering  after  one  revolution,  there  are  no 
orbits  which  for  /z  =  0  reduce  to  those  obtained  in  I,  having  the  period  2vir 
in  T  which  do  not  have  the  period  2ir  also. 

Since  f t ,  £t ,  and  f  are  odd  in  a  and  in  t,  the  orbits  are  symmetrical  with 
respect  to  the  ^-plane,  both  geometrically  and  in  t.  The  two  masses  ml 
and  w2  move  in  the  same  orbit  and  always  remain  180°  apart. 


CHAPTER  XI. 

PERIODIC   ORBITS  OF  INFINITESIMAL  SATELLITES 
AND  INFERIOR  PLANETS. 

179.  Introduction. — This  chapter  is  devoted  to  the  discussion  of  certain 
periodic  orbits  of  the  problem  of  three  bodies  in  which  two  of  the  masses  arc 
finite,  while  that  of  the  third  is  infinitesimal.  The  finite  bodies  are  assumed 
to  revolve  in  circles,  and  the  infinitesimal  body  to  move  in  the  plane  of  their 
motion,  relatively  near  one  or  the  other  of  the  finite  bodies.  The  periodic 
orbits  which  are  obtained  are  those  in  which  the  periods  of  the  solutions  are 
equal  to  the  synodic  periods  of  the  bodies.  The  nearer  the  infinitesimal 
body  is  to  one  of  the  finite  bodies  the  less  its  motion  is  disturbed  by  the  more 
distant  one.  The  orbits  under  discussion  reduce  to  circles  as  the  disturbance 
from  the  more  distant  body  becomes  zero,  and  they  are  therefore  of  the  class 
called  by  Poincare"  "Solutions  de  la  premiere  sorte."* 

The  results  of  this  chapter  are  of  direct  practical  application,  particu- 
larly in  the  Lunar  Theory.  They  are  coextensive  in  this  domain  with  the 
Researches]  of  Hill  and  the  first  memoir  by  Brown. J  When  the  masses  of 
the  two  finite  bodies  have  the  ratio  ten  to  one  the  problem  reduces  to  that 
which  Sir  George  Darwin  treated  by  numerical  processes,  §  and  the  results 
obtained  include  the  orbits  called  by  him  "Satellites  A"  and  "Planets  A." 
Darwin's  "Satellites  B"  and  "Satellites  C"  are  imaginary  for  small  values 
of  the  disturbing  forces  and  belong  to  values  of  the  parameter  for  which  the 
series  obtained  in  this  chapter  do  not  converge.  It  appears  from  the  present 
investigations  that  there  are  three  families  of  satellites  and  of  inferior  planets 
whose  motion  is  direct.  It  follows  that  Darwin's  search  for  them  was  exhaus- 
tive so  far  as  the  satellites  are  concerned;  but  he  found  only  one  family  of 
planets  whose  motion  is  direct.  The  work  of  this  chapter  shows  that  there 
is  an  equal  number  of  retrograde  orbits;  that  is,  three  families  of  real  or 
imaginary  periodic  orbits  around  each  of  the  finite  bodies. 

As  compared  with  previous  work  on  this  subject,  the  methods  of  this 
chapter  are  characterized  by  the  fact  that  the  validity  of  all  the  processes 
employed  is  proved.  They  have  the  merit  of  generality,  not  only  in  showing 
the  number  of  orbits  that  exist,  but  also  in  being  applicable  to  any  ratio  of 
masses  of  the  finite  bodies.  They  have  the  disadvantage  that  the  series  do 
not  converge  for  all  values  of  the  parameter.  The  numerical  processes 

•Let  MUudu  fioweUet  de  la  Mieaniqwi  Ctiette,  vol.  I,  p.  97. 

tAfiMT.  Jovr.  MaOttmatict,  vol.  I  (1878)  pp.  5-28,  129-147,  245-280,  and  Hill's  Collected  Work*,  vol.  I. 

lAmer.  Jour,  of  Mathematics,  vol.  XIV  (1892),  pp.  141-150. 

\Acta  Matkematiea,  vol.  XXI  (1897),  pp.  99-242;  UaOtematitche  Aruuden,  vol.  LI  (1898-9),  pp.  523-583. 


358  PERIODIC    ORBITS. 

employed  by  Darwin  are  easily  applicable  to  the  cases  where  the  series  fail. 
Instead  of  arranging  the  solutions  as  Fourier  series,  as  was  done  by  Hill  and 
as  has  been  customary  among  astronomers,  power  series  are  employed,  and 
the  work  has  the  simplicity  which  is  characteristic  of  processes  involving 
power  series.  For  example,  every  coefficient  is  determined  by  a  single  step 
and  is  modified  by  no  subsequent  operations. 

1 80.  The  Differential  Equations. — Let  m1  and  w2  represent  the  masses  and 
take  the  origin  at  m^  for  the  determination  of  the  motion  of  the  infinitesimal 
body.  Then  the  differential  equations  of  motion  in  polar  coordinates  are 


Pw>  fl  U 

u               /     /\  0     I       A/    it vi  79             I/  Ly                                        //     I     f-\     t      t            79 

M"   __  ,yi  (  ai'    \  ^      1 *  — -    J>*  f¥V)      ^—^—,                                     V*  41      ^1^    x'V*     4)        —    i1^  'VM 

/               I    \V     I  ~~  A/     //to_          •                           /f        |     ^.  /      I/       "™*  IV     I/ Vn 

N          '            '                 rt*Z  *      ,3**      *                                                                                                                                    * 


r2  3r  '  3z;  ' 

r/_  _  1  rcosit^F) 


(1) 


[^2-2r-^cos(t;-y)+r2]}  ^2 

where  r  and  t>  are  the  polar  coordinates  of  infinitesimal  body,  and  R  and  y 
are  the  polar  coordinates  of  mt . 

In  the  present  discussion  those  orbits  are  considered  in  which  r  is  small 
relatively  to  R.  In  this  case  U  is  expansible  as  a  converging  power  series  in 
r/R,  and  equations  (1)  become 


(2) 


It  was  assumed  in  the  beginning  that  the  relative  motion  of  m^  and  ra, 
is  circular.  Hence  we  have,  from  the  two-body  problem, 

Q,  (3) 

where  N  is  the  angular  velocity  of  the  relative  motion  of  the  finite  bodies. 

When  the  right  members  of  the  second  and  third  equations  of  (2)  are 
put  equal  to  zero,  that  is,  when  the  infinitesimal  body  is  supposed  to  revolve 
about  ml  without  being  disturbed  by  ra2 ,  they  have  the  particular  solution 

r  =  a  =  constant,        v= — ^  (t  —  4)  =n(t— Q,  (4) 

U 

where  n  is  the  angular  velocity  of  the  infinitesimal  body  with  respect  to  mt . 
In  the  periodic  solutions  of  (2)  when  the  right  members  are  included,  the 
mean  angular  velocity  will  be  kept  equal  to  n  and  a  will  be  defined  by  the 
equation 

n2a'  =  A;2m1.  (5) 


INHMIKSIMAL   SATELLITES   AND   INFERIOR   PLANETS. 

Since  ///,  is  given,  a  has  three  determinations,  two  of  which  are  complex.  In 
the  physical  two-body  problem  there  is  interest  only  in  the  real  value  of  a, 
and  it  is  immaterial  whether  corn  is  regarded  as  given.  It  will  be  observed, 
however,  from  the  purely  astronomical  point  of  view,  that  n  can  be  determined 
by  observation  much  more  accurately  than  a,  because  the  former  is  an  angular 
variable  and  any  error  in  its  determination  causes  the  theory  to  deviate 
secularly  from  the  observations;  while  the  latter  is  a  linear  variable  and  the 
discrepancies  which  arise  from  errors  in  its  determination  do  not  accumu- 
late. But  in  the  three-body  problem  all  three  values  of  a  must  be  included 
in  determining  orbits  whose  period  is  defined  by  n,  because  all  the  orbits 
may  become  real  for  certain  values  of  the  parameters  which  occur  in  the 
right  members  of  the  differential  equations.  In  fact,  Darwin  found  three  real 
orbits  for  certain  values  of  the  Jacobian  constant.* 

In  the  Lunar  Theory,  as  developed  by  de  Pont4coulant,  for  example,  the 
solutions  were  forced  into  the  trigonometric  form  by  various  artifices.  But 
it  will  be  noticed  that  the  method  of  procedure  was  the  opposite  of  that 
adopted  here,  so  far  as  comparison  can  be  made,  in  that  the  mean  motion 
was  continually  modified  by  de  Pont^coulant  in  order  to  preserve  the 
trigonometric  form;  while  here  the  mean  motion  is  regarded  as  being  given 
arbitrarily  in  advance  by  the  observations  or  otherwise,  and  it  is  kept  fixed. 

New  quantities  p,  w,  r,  and  m  will  be  introduced  by  the  equations 


(6) 


For  brevity  a/  A  will  be  written  in  place  of  its  expression  as  a  power  scries. 
As  a  result  of  these  transformations,  the  last  two  equations  of  (2)  become 


!j(l+p)[3cos(T+uO+5cos3(T+u>)]+ 


(7) 


where  the  dots  over  p  and  w  indicate  derivatives  with  respect  to  T.  These 
equations  are  valid  for  the  determination  of  the  motion  of  the  infinitesimal 
body  as  long  as  a(l+p)  is  less  than  A. 

•Ada  MaOiematica,  vol.  XXI  (1807),  pp.  W-242. 


360  PERIODIC    ORBITS. 

It  follows  from  equations  (6)  that  ap  and  w  are  the  deviations  from  uni- 
form circular  motion  due  to  the  right  members  of  the  differential  equations. 
They  are  functions  of  m,  and  the  initial  conditions  are  to  be  determined  so 
that  they  shall  be  periodic  in  T  with  the  period  2ir.  Since  the  right  members 
of  (7)  contain  m2  as  a  factor,  the  periodic  expressions  for  p  and  w  will  contain 
m2  as  a  factor. 

181.  Proof  of  the  Existence  of  the  Periodic  Solutions. — For  m  =  Q 
equations  (7)  admit  the  periodic  solution  p  =  w  =  0.  It  will  now  be  proved 
that  for  m  distinct  from  zero,  but  sufficiently  small,  equations  (7)  admit  a 
periodic  solution  which  has  the  period  2ir  in  T,  and  which  is  expansible  as  a 
power  series  in  m1/3,  vanishing  with  m.  It  is  the  analytic  continuation  of  the 
solution  p  =  w  =  m  =  Q  with  respect  to  m  as  the  parameter.  Since  T  enters 
explicitly  in  the  right  members  of  (7)  in  terms  having  the  period  2ir,  it  follows 
that  the  period  of  the  solution  must  be  2?r,  or  a  multiple  of  2ir. 

Equations  (2)  are  not  altered  if  we  change  the  sign  of  v,  V,  and  t.  It 
easily  follows  from  this  that,  if  v  (0  =  V  (0  =  p'  (<„)  =  0,  the  dependent  vari- 
ables p  and  v  are  even  and  odd  functions  respectively  of  t  — 10 .  An  orbit  in 
which  these  conditions  are  satisfied  is  symmetrical  with  respect  to  a  line 
always  passing  through  m1  and  w2 .  Such  an  orbit  will  be  called  symmetrical. 
Expressed  in  the  variables  of  (7),  p  and  w  are  respectively  even  and  odd 
functions  of  T  in  the  case  of  symmetrical  orbits;  and  w(ty  =p(0)  =0. 

The  existence  of  symmetrical  periodic  orbits  will  be  established,  and  then 
it  will  be  shown,  in  connection  with  the  construction  of  the  solutions,  that  the 
condition  that  the  solutions  are  periodic  implies  that  they  are  also  sym- 
metrical. It  will  follow  from  this  that  all  of  the  periodic  orbits  of  the  type 
under  consideration  are  symmetrical. 

Suppose  that  the  initial  conditions  are 

(8) 


If  the  right  members  of  equations  (7)  are  neglected  and  the  transformation 


is  made,  then  equations  (7)  reduce  to  the  ordinary  polar  equations  of  the 
two-body  problem  for  the  motion  of  the  infinitesimal  body  with  respect 
to  ml  .  In  this  two-body  problem  with  the  initial  conditions  (8),  it  is  found 
that  aa  is  the  increment  to  the  semi-axis  a,  and  e  is  the  eccentricity  of 
the  orbit  of  the  infinitesimal  body.  Since  the  properties  of  the  solution  of  the 
two-body  problem  in  terms  of  the  major  semi-axis  and  eccentricity  are 
known,  we  can  at  once  write  down  the  properties  of  the  solution  of  (7)  in 
terms  of  a  and  e,  so  far  as  they  are  independent  of  the  right  members  of  the 
equations.  It  was  precisely  for  this  reason  that  p  and  w  were  given  the 
peculiar  initial  values  defined  in  (8). 


INHMI1>1\IAL   SATELLITES   AND    INFERIOR    PLANETS.  361 

Equations  (7)  arc  now  integrated  as  |>o\ver  series  in  a,  e,  and  m{/t. 
The  results  have  the  form 

p  =  pl(a,e,m^;r),  w  =  p,  (a,  e,  w1*;  T),  1 

P  =  pt  (a,  e,  w"»;  T),  w  =  p.  (a,  e,  ro1*;  T). 

The  moduli  of  a,  c,  and  m';i  can  be  taken  so  small  that  the  series  converge  while 
T  runs  through  any  finite  range  of  values  starting  from  zero.  The  period 
must  be  a  multiple  of  2*-,  say  2kir,  and  consequently  it  will  be  supposed  that 
these  parameters  have  such  small  moduli  that  (9)  converge  for  O^r^A'ir. 

It  follows  from  the  symmetry  of  the  orbits  under  the  initial  conditions 
(8),  that  if,  at  r  =  kir,  the  three  bodies  are  in  a  line,  and  if  the  infinitesimal  body 
is  crossing  perpendicularly  the  rotating  line  which  joins  wr  and  mt,  then  the 
orbit  necessarily  re-enters  at  2kr,  and  the  motion  is  periodic  with  the  period 
2kir.  Conversely,  if,  at  r  =  0,  the  orbit  crosses  perpendicularly  the  rotating 
line  which  joins  m,  and  mt,  and  if  the  motion  is  periodic  with  the  period  2ir, 
then  the  orbit  at  r  =  kir  necessarily  crosses  perpendicularly  the  rotating  line 
which  joins  »i,  and  w,.  Therefore,  necessary  and  sufficient  conditions  for 
the  existence  of  symmetrical  periodic  solutions  having  the  period  2k*  are 

p,(a,  e,  m"»;  *«•)  =0,  p,(o,  e,  m*;  kw)  =  0.  (10) 

Since  pt  =  pt  =  0  at  r  =  0,  it  follows  that  equations  (10)  are  identically  satis- 
fied by  a  =  e  =  wl/1  =  0.  The  problem  of  solving  them  for  a  and  e  as  power  series 
in  ml/l,  vanishing  for  m  =  0,  will  now  be  considered.  All  the  terms  of  the  solu- 
tions which  do  not  carry  w*  as  a  factor  are  obtained  from  the  solutions  of  the 
left  members  of  (7)  set  equal  to  zero.  Since  these  terms  belong  to  the  two- 
body  problem  equations  (9)  become,  when  these  terms  are  explicitly  exhibited, 


(11) 


where  /19x 

"(1+a)" 

The  series  9,  ,  .  .  .  ,  qt  depend  upon  the  right  members  of  the  differential  equa- 
tions (7).    All  terms  in  the  [  ]  which  are  not  written  contain  e1  as  a  factor. 
The  conditions,  (10),  for  the  existence  of  symmetrical  periodic  solutions 
become  as  a  consequence  of  (11) 


-  1)+  •  •  - 

+  •  •  •  J-l-m1  &(a,  e,  m1";^, 
(l+m)T+»T-r-2esinjT-f  •  •  •+mtg,(o,e,  m^;  T), 


t(a,e,mtn;kir)  =  0, 

[(13) 
R(*»)  -  L-(l-H»)*r-h*r-f ae«na*»4-  •  • 


362  PERIODIC   ORBITS. 

where  all  the  unwritten  terms  in  the  [  ]  carry  e2  as  a  factor  and  are  linear 
functions  of  sines  of  multiples  of  vkir.  On  substituting  the  value  of  v  from 
(12),  it  is  found  that 


r  =   -        sn    m-a      •  • 

where  j  is  an  integer.  Every  term  in  the  [  ]  contains  either  m  or  a  as  a  factor; 
therefore  every  term  of  the  second  equation  of  (13)  contains  either  m  or  a 
as  a  factor.  The  coefficient  of  a  to  the  first  power  is  —3/2  kir,  which  is 
distinct  from  zero;  therefore  the  second  equation  of  (13)  can  be  solved  for 
o  as  a  power  series  in  m  and  e,  vanishing  with  m  =  0.  The  term  of  lowest 
degree  in  m  alone  is  the  second,  and  its  coefficient  depends  upon  the  right 
member  of  the  differential  equations  (7).  Therefore  the  solution  of  the 
second  equation  for  a  carries  m2  as  a  factor,  and  has  the  form 

a  =  m*p(m1/3,  e).  (14) 

Suppose  a  is  eliminated  from  the  first  of  (13)  by  means  of  (14).  After 
the  elimination,  a  factor  m  can  be  divided  out.  The  result  then  contains 
a  term  in  e  alone  to  the  first  degree,  and  its  coefficient  is  (  —  1)*.  Therefore 
the  resulting  equation  can  be  solved  for  e  as  a  power  series  in  mt/3,  vanishing 

with  m  =  0,  of  the  form 

l/3).  (15) 


As  a  matter  of  fact,  the  expression  for  e  contains  m2  as  a  factor,  as  can 
easily  be  shown.  Suppose  equations  (7)  are  integrated  as  power  series  in  m1/3, 
and  let  the  initial  conditions  be  p  =  P  =  w  -  w  =  0,  in  order  to  get  the  terms 
which  are  independent  of  a  and  e.  The  series  will  have  the  form 

p  =  p1ra+p2ra2+p3ra8/3+  •  •  •  >         w  = 
The  explicit  result  of  the  integration  is 


r-l+2cos-r-cos2r1, 

1- 


From  these  equations  it  follows  that  p2  (kir)  has  m2  as  a  factor,  but  that  it 
does  not  have  m3  as  a  factor.  Therefore  the  expression  for  e  as  a  power 
series  in  w1/3  contains  m2  as  a  factor.  Then  (15)  and  (14)  together  give 

o  =  mtP1(w1A),  e  =  m2P8(m1/3),  (16) 


INFINITESIMAL   SATELLITES   AND   INFEHIoK    I'LANETB. 

where  I\  and  Pt  arc  power  series  in  m1'3  which  converge  for  the  modulus 
of  m  sufficiently  small.  Therefore  t he  symmetrical  periodic  orbits  e\i>t.and 
equations  (16)  and  (8)  give  the  initial  values  of  the  dependent  variables  in 
terms  of  the  parameter  m1"  which  is  defined,  except  for  the  cube  root  of 
unity,  by  the  data  of  the  problem  and  the  third  equation  of  (6). 

182.  Properties  of  the  Periodic  Solutions. — The  periodic  orbits  whose 
existence  lias  been  proved  re-enter  after  the  period  2for,  where  k  is  any 
integer.  Those  orbits  for  which  k  is  greater  than  unity  include  those  for 
which  k  equals  unity.  Since,  according  to  the  discussion  which  has  just 
been  made,  the  number  of  periodic  orbits  is  the  same  for  all  values  of  k,  it 
follows  that  the  period  of  the  solutions  is  2r.  When  w  =  0  the  infinitesimal 
body  makes  a  revolution  in  2*-,  and  m  can  be  taken  so  small  that  the  orbit 
is  as  near  this  undisturbed  orbit  as  may  be  desired.  Therefore  a  synodic 
revolution  is  made  in  2ir  for  all  |m|  sufficiently  small. 

If  the  expressions  for  a  and  e  given  in  (16)  are  substituted  in  (9),  the 
result  becomes  »  » 

p=£p,(T)m'",         w-  Z^Wro'*,  (17) 

where  the  summation  starts  with  .7  =  6,  because  the  expressions  for  a  and  e 
have  m1  as  a  factor,  and  p  and  w  have  no  terms  in  m  alone  of  degree  less  than 
the  second. 

The  p  and  w  are  periodic  with  the  period  2*  for  \m\  sufficiently  small 
because  the  conditions  for  periodicity  have  been  satisfied.  Therefore 

I  Pj  (T + 2  T)ro'/J  =  S  p,  (T)ro"»,  ^ w,  (r + 2 T)m'/J  =  2  w,  (r)m'" ; 

whence 

p,(T+2T)  =  p,(r),        Wj(T+2^=Wj(r')         (j-6, ...«).     (18) 

Since  p(0)  =  u?(0)  =0,  it  follows  that 

2  p,  (0)  m"1  =  0,  2  w,  (0)m'/'  =  0 ; 

;-•  /-• 

whence 

p/0)=0,        t^(0)=0  tf-6,  ...oo).  (19) 

If  the  orbit  of  the  infinitesimal  body  is  retrograde,  n  is  negative  and  m 

has  the  definition 

m=—  N/(n+N) 

for  a  given  sidereal  period.  Therefore,  for  a  given  numerical  value  of  n,  the 
parameter  m  is  smaller  in  retrograde  motion  than  it  is  in  direct  motion.  For 
a  given  sidereal  period  the  deviations  from  circular  motion  are  less  in  the 
retrograde  orbits  than  they  are  in  the  direct.  The  physical  reason  is  that 
the  disturbance  of  the  motion  of  the  infinitesimal  body  by  m,  is  greatest 
when  the  three  bodies  are  in  a  line,  as  can  be  seen  from  (7)  or  by  graphically 
resolving  the  disturbing  acceleration;  and  in  retrograde  motion  this  approxi- 
mate condition  lasts  a  shorter  time  than  in  direct  motion. 


364  PERIODIC    ORBITS. 

DIRECT  CONSTRUCTION  OF  THE  PERIODIC  SOLUTIONS. 

183.  General  Considerations.-  —  It  has  been  proved  that  equations  (7) 
have  solutions  of  the  form  (17)  which  satisfy  (18)  and  (19).  The  solutions 
are  in  ml/3  only  because  a/  A  is  a  series  in  m1/3,  given  explicitly  in  (6).  The 
expression  for  a/A  can  be  modified  by  writing 

±  =  Mm,  (20) 

where  M  is  to  be  regarded  in  the  analysis  as  a  constant  independent  of  m. 
This  amounts  to  a  generalization  of  the  m  as  it  appears  in  certain  places  in 
the  last  equation  of  (6).  The  particular  transformation  (20)  is  made  in 
order  that  the  right  members  of  (7)  shall  be  in  integral  powers  of  m.  The 
proof  of  the  existence  of  the  periodic  solutions  can  be  made  precisely  as 
before,  because  the  transformation  (20)  affects  only  the  higher  terms  which 
were  not  explicitly  used.  While  there  is  nothing  essential*  in  the  trans- 
formation, it  will  be  made  for  the  sake  of  convenience,  after  which  equations 
(7)  become 


(21) 


where 


In  the  right  member  of  the  first  equation  of  (21)  the  coefficient  of  m'  is 
a  sum  of  cosines  of  integral  multiples  of  T+W,  the  highest  multiple  being  j; 
the  coefficient  of  mj  in  the  right  member  of  the  second  equation  is  a  sum  of 
sines  of  integral  multiples  of  (T+W),  the  highest  multiple  being  j. 

In  a  closed  orbit  around  ml  there  are  two  points  at  which  w  is  zero. 
The  arbitrary  £„  will  be  so  determined  that  w(0)=0.  The  first  condition 
of  (8)  will  not  be  imposed  in  advance,  and  it  will  be  shown  that  it  is  a  conse- 
quence of  the  others.  It  will  follow  from  this  that  all  of  the  periodic  solutions 
of  the  type  under  consideration  are  symmetrical.  Equations  (21)  will  there- 
fore be  integrated  in  the  form 

CO  00 

p  =  S  Pi  m',         w=2  w}m!  (22) 


,/=2 


subject  to  the  conditions  (18)  and  the  second  of  (19). 

*A  different  transformation  was  made  in  Transactions  of  the  American  Mathematical  Society,  vol.  VII, 
(1906),  p.  542. 


INFINITESIMAL   8ATKI.I.ITKS   AND   INFERIOR   PLANETS. 

184.  Coefficients  of  m-.     'l'\n-»-  term-  an-  <l<Tm<><l  l>y  the  (Munitions 
P1-3pJ-2»-,  =  |»;(l+3cos2T),         tPt+2^=  -|rj8in2r, 

-  A 

the  general  solution  of  which  is 

P,  =  \  i\ + 2  c{" + c?  cos  T  -f  cf  sin  T  -  jj  cos  2r , 

"':  =  cr- (7j+3cItt))r-2cIfI>8inT+2cia)cosT+ TJ  sin2r, 


(23) 


when-  r,-',  .  .  .  ,  c™  are  the  constants  of  integration.     By  conditions  (18)  and 
tin-  second  of  (19),  it  follows  that 


(24) 
Therefore  the  solution  (23)  becomes 


p,=  — 


wt  =  -  2  cj8  -  2c®  sin  r  +  2^  cos  r  +  ij-ij  sin  2r, 

O 

where  ci"  and  cf  are  constants  which  remain  so  far  undetermined. 


(25) 


185.  Coefficients  of  m'. — The  differential  equations  which  define  the 
t  en i is  of  the  third  degree  in  m  are 


ivt+2pt=  -2p,- 1 

=  f  2cf  - 1  Mrj)  sin  T  -  2  c?>  cos  T  -  4 17  sin  2r  -  ^  Mn  sin  3r. 

\  o  /  8 

From  the  second  of  these  equations  it  is  found  that 
which  substituted  in  the  first  gives 


366  PERIODIC    ORBITS. 

In  order  that  the  solution  of  this  equation  shall  be  periodic  the  coefficients 
of  COST  and  SUIT  must  be  zero;  whence 


<?-0,  (29) 

after  which  the  general  solution  of  (28)  becomes 

P3  =  -  77 + 2  cf  + cf  cos  7 + cf  sin  7  -  J  77  cos  27  -  !  MT?  cos  37,  (30) 

where  cf ,  cf ,  and  cf  are  undetermined  constants.     The  result  of  substi- 
tuting this  expression  in  (27)  is 

:f +  -r  Mr]  Jcos7— 2cf  sin  T+ —77  cos  27+  —Af77cos3r.    (31) 

In  order  that  the  solution  of  this  equation  shall  be  periodic  the  right  member 
must  contain  no  constant  term;  whence 

„«>  _  J2  /->9\ 

Cj    —     77.  v'J^/ 

With  this  value  of  cf  it  is  found,  upon  integrating  (31)  and  imposing  the 
second  condition  of  (19),  that,  so  far  as  the  computation  has  been  made, 

1  15  TI  ,r  15  it  f       •  !!• 

P2=  — -77  +  — M77COS7  — 7jcos27,          w>2  =  —  —  A/??  sin  7+--77sm27, 


6 

P3= +-77 +cf  cos7+cf  sin7- -7JCOS27  -  —  M77cos37,  [(33) 

w3=- 2cf  +  2cf  cos  7  -  (  2 cf  + 1 M-n )  sin  7  +  ^ 77 sin 2r  +  ^¥77  sin 37, 
where  cf  and  cf  are  constants  which  are  as  yet  undetermined. 

186.  Coefficients  of  m4. — The  integration  will  be  carried  one  step 
further  and  then  the  induction  to  the  general  term  of  the  solution  will  be 
made.  The  differential  equations  which  define  the  coefficients  of  m4  are 


-  3  w2  sin  2r  -f  ^  Af2r?  [9 + 20  cos  2r  +  35  cos  4r] , 

(34) 

•  .«^» 

'4  =  —  p2  w2~  2  p3  —  2p2  w2  —  -  p2  sin  2r  —  3ir2  cos  2r 


IMIMI1MMAL   SATELLITES   AND    INFERIOR   PLANETS. 

Upon  developing  the  explicit  values  of  the  right  members  of  (34)  by 

means  of  (33.),  it  U  found  that 


-f  {  M  «,  ]  cos  2r  +  |Mn  cos  3r  +  [I  »,'+  f6  M  V,]  cos  4r, 


(35) 


The  first  integral  of  the  second  of  these  equations  is 


(36) 


where  c{°  is  an  undetermined  constant.     Then,  on  substituting  this  value 
of  w^  ,  the  first  equation  of  (35)  becomes 


(37) 


In  order  that  the  solution  of  (37)  shall  be  periodic,  the  conditions 

ci»  =  fM,'-|M^  cr=0  (38) 

must  be  satisfied.     Then  its  general  solution  becomes 


675  M t  t  _i_  JL 
512  M  '   "h!6J 


(39) 


where  of,  cJ4),  and  ci"  are  as  yet  undetermined  constants. 


368  PERIODIC    ORBITS. 

If  (39)  is  substituted  in  (36),  it  is  found,  by  using  (38),  that 


(40) 


The  periodicity  condition  determines  cj*  by  the  equation 


Then  the  integral  of  (40)  satisfying  the  second  condition  of  (19)  is 


(42) 


The  results  so  far  obtained  are 
P2=  — 


t=-  2c<4)  +2cf  cos  r  -  [2c<4)  +  1|  Mr;2  -  ^  Af  1,]  sin  r 

AfV+|AP,]8in2r 


(43) 


where  rj=  -  ^—  -  ,  and  where  Cj4>  and  c"'  are  so  far  undetermined. 

~~ 


IXFIXITF.SIM.M.    S-ATF.I.I.MT.s    AND    INFERIOR    PLANETS.  369 

It  is  observed  that,  so  far  as  the  variables  are  completely  determined,  the 
p,  and  »•  are  sums  df  cosines  ;iiid  sine-  respectively  of  integral  multiples  of  r, 
the  highest  multiple  being  7.  At  the  j'h  step  of  the  intejiration  one  of  the 
four  arl  lit  rary  constants  which  arise  at  that  step  is  determined  by  the  perio- 
dicity condition  on  the  w,,  and  another  by  the  initial  condition  on  the  w,. 
The  other  two  constants  remain  undetermined  until  the  next  step,  but  two 
which  arose  at  the  preceding  step  are  determined  by  the  periodicity  condition 
on  p,.  It  will  be  shown  that  these  properties  are  general. 

187.  Induction  to  the  General  Step  of  the  Integration.  Suppose 
P5,  .  .  .  ,  p.-,;  M'J,  .  .  .  ,  wn^i  have  been  computed  and  have  the  properties 
expressed  in  the  following  equations: 

p,  =  C  +  a'l"cosT+a?cos2T+  •  •  •  +CL?COSJT        (j=2,  .  .  .  ,  n-2), 
w,=         /3I(/)sinT+/310)sin2T+  •  •  •  +fftanjr       (j=2,  ....  n-2), 
P.-,=  +c;-1)sinT+o;"-1)+ci-"cosT+a;-l)cos2T+  •    •  +a(::II)cos(n-l)r,    (44) 


where  the  a£°,  0J0,  and  &{"~u  are  known  constants,  and  cj""'  and  cj"~"  are 
undetermined  constants. 

In  writing  the  differential  equations  which  define  p,  and  IP,  all  unknown 
<  luant  it  ies  will  be  given  explicitly.  The  terms  involving  these  undetermined 
coefficients  are  the  same  at  every  step.  It  is  found  from  equations  (7)  that 
the  coefficients  of  ra"  are  defined  by 

p.-3p.-2w.=  +2r;-'>cosT+2crl)sinT+P.(p,,  w,,  w,;  r),     } 

I    (45) 
p/,  p,,  w,,  w;  T).J 


where  P.  and  Qn  are  polynomials  in  p,,  p,,  w,,  and  w,  (j  =  2,  .  .  .  ,  n-2)  and 
the  known  parts  of  p._i ,  p._i ,  w.-t ,  and  wu-\,  and  where  T  enters  in  the 
coefficients  only  in  sines  and  cosines. 

It  follows  from  (7)  that,  aside  from  numerical  coefficients,  P.  has  terms 
of  the  types 

Pi"  =  ph  Wh  0',  +h  =  n,  or  n  - 1 ) , 

PI*  =  ph  Wh  tbA  (Ji+h+h  =  »), 

Pi1)  =  p*1'  •   •   •  p*'  (kji+  '  '  '  +krjr  =  n,  n  — l,orn— 2), 

Pi4'  =  M'p*'t  •  •  •  p};u£'  •  •  •  wJji'i'JTJj     O'-0, 1,  ....  n-2;  *;,+  •••+! 
t'^.7+2;  j+kJt+  •  •  •  +k,j,+\pi+  •  •  •  +\p 

If  X,+  •  •  •  +XMiseven,  the  term  is  multiplied  by  costV;  and  if  X,+ 

is  odd,  the  term  is  multiplied  by  sinir.    The  terms  Pi",  Pf,  and  P?  come 

from  the  left  member  of  (7),  and  P?  comes  from  the  right  member. 


370  PERIODIC   ORBITS. 

It  follows  at  once  from  (44)  and  the  conditions  on  the  j,  and  kt  that 
P»\  Pn\  and  P™  are  sums  of  cosines  of  integral  multiples  r,  the  highest 
multiple  being  n  at  most.  If  Xt+  •  •  •  +XM  is  even,  the  product  w%  •  •  •  w^  is  a 
sum  of  cosine  terms,  and  it  follows  therefore  that  in  this  case  PM4)  is  a  sum  of 
cosines  of  integral  multiples  of  r.  The  highest  multiple  of  T  is 


which  becomes,  as  a  consequence  of  the  relations  to  which  the  exponents 
and  subscripts  are  subject, 


If  A!+  •  •  •  +XM  is  odd,  the  product  w%  •  •  •  wfy  is  a  sum  of  sines  of 
integral  multiples  of  T.  Therefore,  in  this  case  also,  P™  is  a  sum  of  cosines 
of  integral  multiples  of  T;  and  it  is  shown,  precisely  as  before,  that  the  highest 
multiple  is  n.  Hence  the  general  conclusion  is  that  PB  is  a  sum  of  cosines 
of  integral  multiples  of  T,  the  highest  multiple  being  n. 

By  a  similar  discussion  it  can  be  proved  that  Qn  is  a  sum  of  sines  of 
integral  multiples  of  T,  the  highest  being  n.  Hence  equations  (45)  can  be 
written  in  the  form 


(46) 


where  the  A™  and  the  B™  Q'  =  0,  .  .  .  ,  ri)  are  known  constants. 
The  first  integral  of  the  second  equation  of  (46)  is 

wn  =  -  2  pn+c?  -  2  c'"-"  sin  r  -  [2  c?-°  +B?]  cos  T  - 1 B™  cos  2r 

j  (47) 

_    ... $£"  Cos  nr, 

n 
where  cJB)  is  an  undetermined  constant. 

On  substituting  equation  (47)  in  the  first  of  (46),  it  is  found  that 


cosnr, 
wa+2Pn=-2  cf-l>  cos r+[2 c'"-"  +B™]  sin  r 

+5Bn)sinnr, 


2 

D(n)  1  cos  nr. 


(48) 

In  order  that  the  solution  of  this  equation  shall  be  periodic,  the  conditions 

(49) 


must  be  imposed.     They  uniquely  determine  the  constants  cJ"~J  and  cj""", 
which  remained  undetermined  at  the  preceding  step  of  the  integration. 


INKIMCKSIMAL   SATKLUTKS   AND    INKKUloK    PLANETS.  371 

After  e(iuations  (49)  arc  fulfilled,  the  general  solution  of  (48)  is  of  the  form 

p.  =  ci")siiiT  +  ai">+ci")cosT  +  ai">co82r+   •  •  •   +  ai,"  cos  HT,          (50) 
where  , •/'  and  c™  arc  arbitrary  constants,  and  where 

a<->-       _L_r4<->- *  »<•>-)-       \JA?-2BT]  <51> 

tt;    •          -j .  \_    I        4     '  J  ~   1(1* i  i"  U~*>  •  •  •  i  "/• 

If  e<iuations  (49),  (50),  and  (51)  are  substituted  in  (47),  the  result  is 


(52) 


In  order  that  the  solution  of  this  equation  shall  be  periodic,  its  right  member 
must  contain  no  constant  term;  whence 


(53) 

Then  the  integral  of  (52)  satisfying  the  condition  w?.(0)  =  0  is 


(54) 


i^)]8in2r  -----  i  [2  <+}*;'  >in 


The  results  obtained  at  this  step  are 

•  •  +<)cosjr 

+  •  •  •   +a|>11)cosnr, 


+  •  •  •  +  ft"'  sin  TIT, 


(55) 


where  ci"  and  cjm)  are  as  yet  undetermined  constants. 


372  PERIODIC   ORBITS. 

Since  the  results  expressed  in  the  first  two  equations  of  this  set  are 
identical  in  properties  with  the  equations  (44),  with  which  the  discussion  of 
the  general  step  was  started,  and  since  the  properties  of  (44)  were  fulfilled 
for  the  subscripts  2,  3,  and  4,  it  follows  that  the  induction  is  complete.  The 
process  of  integration  can  be  carried  as  far  as  may  be  desired. 

The  hypotheses  under  which  the  discussion  has  been  made  are  that  the 
solutions  are  periodic  and  that  w(Q)  =  0.  Solutions  satisfying  these  properties 
and  p(0)  =0  were  known  to  exist  from  the  existence  discussion,  and  therefore 
they  could  certainly  be  found  because  the  assumed  properties  are  included 
in  those  of  the  symmetrical  orbits.  It  appears  in  the  construction  that 
the  hypotheses  adopted  imply  also  that  p(0)  =0.  Therefore  all  periodic 
orbits  of  the  type  under  discussion  which  are  expansible  as  power  series  in  m 
are  symmetrical  orbits.  It  can  be  shown  by  direct  consideration  of  the  series 
that  there  are  no  others  expansible  in  any  fractional  powers  of  m. 

188.  Application  of  Jacobi's  Integral. — The  differential  equations  admit 
Jacobi's  integral,  of  which  no  use  has  yet  been  made,  and  it  is  the  only 
integral  not  involving  the  independent  variable.  Upon  transforming  the 
integral  to  the  variables  used  in  this  chapter,  it  is  found  without  difficulty 
that  its  explicit  form  is 


(56) 


where  C  is  the  constant  of  integration.  It  will  be  shown  that  this  integral 
can  be  used  as  a  searching  test  on  the  accuracy  of  the  computations  of  the 
solutions,  or  to  replace  the  second  differential  equation  of  (7). 

Since  the  periodic  solutions  are  developable  as  power  series  in  m,  the 
integral  can  be  expanded  as  a  power  series  in  m  and  written  in  the  form 

F0+Fl(pi,pj,w3,w1;r)m+  •  •  •  +Fn(p1,'p],w1,wj;T}mn+  •  •  •  =C.     (57) 

In  the  Fn  the  highest  value  of  j  is  n.  Since  the  integral  converges  for  all 
\m\  sufficiently  small,  each  Fn  separately  is  constant,  and  therefore 

F«(PJ,  pi,  Wj,  w};  r}  =  Ca.  (58) 

It  follows  from  the  form  of  (55)  and  (56)  that  Fn  is  a  sum  of  cosines  of 
integral  multiples  of  r.  In  Fn  the  sum  of  the  products  of  the  exponents  and 
subscripts  of  the  factors  of  any  term  not  involving  COSJT  or  sinjr  can  not 
exceed  n;  and  in  any  term  involving  COSJT  or  sinjr  the  sum  can  not  exceed 
n—i.  Therefore  the  highest  multiple  of  r  in  Fn  is  n,  and  (58)  can  be  written 
in  the  form 

•  •  •  +7fcos;V+  •  •  •  +7™  cos  TIT  =  (7,,.         (59) 


1+P 


INFINITESIMAL   SATELLITES   AND    INKKIUoK   PLANETS.  373 

Since  this  relation  is  an  identity  in  T,  it  follows  that 

T.W-C.,         7«-0  0-1,  ----  n).      (60) 


relations  are  functions  of  a®,  .  .  .  ,  aj°;  /3f,  .  .  .  ,  0*',  and  their 
fulfillment  forj=  1,  .  .  .  ,  n  serves  as  a  thorough  check  on  the  expansion  of 
t/  and  on  all  of  the  computations. 

It  will  now  be  shown  how  the  relations  (60)  can  be  used  in  place  of 
the  second  equation  of  (7).  If  (56)  is  expanded  as  a  power  series  in  m, 
it  is  found  that 

F.  =  4  p.+2w.+4  pb.,+0.  (p,,  p,,  »„  tb,;  T),  (61) 

where  G,  is  a  polynomial  in  ph  p,,  wif  and  w,  and  involves  r  only  in  sines 
and  cosines.  Moreover,  the  greatest  value  of  j  in  G,  is  n—  2.  Suppose 
that  p,,  .  .  .  ,  p._,;  u>lf  .  .  .  ,  w>._,  are  entirely  known,  and  that  p._,  and  to,_i 
are  known  except  for  the  undetermined  coefficients  cj"~";  it  will  be  shown 
that  equations  (60)  and  the  third  and  fourth  equations  of  (55)  define  the  aj" 
and  /3J"  uniquely. 

It  follows  from  the  properties  of  Fu  and  equation  (61)  that  this  func- 
tion can  be  written  in  the  form 


Consequently  equations  (60)  become 

(62) 


7,w  =  4ar+2^r+CT  =  0 
It  follows  from  ccjuations  (55)  that 


Upon  comparing  equations  (62)  and  (63),  it  is  found  that 


(63) 


2BJ*-/Cy      (n  =  2,  .  .  .  oo  ;  j-1,  .  .  .  ,  n).  (64) 

Therefore  the  third  to  the  seventh  equations  of  (55)  can  be  written 


_     -    ,     -,  _     _-  . 

1-  - 


(65) 


374  PERIODIC    ORBITS. 

These  equations  express  the  coefficients,  which  are  determined  at  this  step  of 
the  integration,  uniquely  in  terms  of  constants  which  depend  only  upon  the 
first  equation  of  (7)  and  upon  the  integral.  In  practical  computation  it  is 
more  convenient  to  make  the  determination  of  the  coefficients  depend  upon 
the  A™  and  C™  than  upon  the  A™  and  B™,  for  the  former  have  many 
terms  in  common,  except  for  numerical  multipliers,  and  both  are  coefficients 
of  cosine  series,  which  are  easier  to  check  than  are  the  sine  series  on  which 
the  B™  depend.  But  the  chances  of  error  in  lengthy  computations  are  so 
great  that  if  the  developments  are  to  be  made  to  high  powers  of  m,  the  only 
safe  method  is  to  use  both  the  second  equation  of  (7)  and  the  integral,  or, 
what  is  the  same  thing,  to  secure  the  fulfillment  of  equations  (64). 

In  order  to  illustrate  the  process  the  expression  for  F^  will  be  developed. 
It  is  found  from  equation  (56)  that 


(66) 


Upon  developing  the  right  member  explicitly  by  means  of  the  first  four 
equations  of  (43),  it  is  found  that 


sin  2r+^M*r)  [9+20  cos2r+  35  cos4r] . 


cos  2r  +     4a»  +  6#>  +  f 


It  is  found  in  the  notation  of  (62),  and  by  comparing  with  the  right  member 
of  the  second  equation  of  (35),  that 


(67) 


exactly  fulfilling  equations  (64). 


INFINITESIMAL   SATELLITES   AND   INFERIOR   PLANETS.  375 

189.  The  Solutions  as  Functions  of  the  Jacobian  Constant. — It  follows 
from  equation  (57)  that  when  the  periodic  solution  is  given,  the  constant  C  is 
uniquely  denned.  The  relation  between  C  and  the  constant  of  the  Jacobian 
integral,  when  it  is  expressed  in  terms  of  the  variables  in  more  ordinary 
use,  will  ho  found.  If  the  origin  is  taken  at  the  center  of  gravity  of  the 
system,  the  differential  equations  of  motion  in  rectangular  coordinates  are 


t  _  dU  f         OU  jr        K  Jill    | 

—  "••  |  ^A     -=  '-  J  fy    ^S        ~  '    "•    •  f • 

dx  aw  n 


(68) 


These  equations  admit  the  integral 

x*+y'*-2N(xy'-yx')  =  2U-Ct,  (69) 

where  C0  is  the  constant  of  the  Jacobian  integral  and  Af  is  denned  in  (3).    The 
relation  between  C0  and  C  of  equation  (56)  is  required. 

The  variables  x  and  y  are  expressed  in  terms  of  the  polar  coordinates, 
r  and  v  of  (1),  by  the  equations 

z  =  rcosi; —       —  AcosNt,  y  =  rs\nv—-        —  AsinM; 

from  which  it  follows  that 

z"+ y'*  -  2N(xy'  -  yx')  =  r"+rV'  -  2Nr*v' 


2m1 


AJVrcoB(»-M)  -  mJAW. 


(70) 


Upon  making  the  transformations  (6)  and  referring  to  (3)  and  (5),  it  is  easily 
found  that 


,  ,         »  = 
r,          r,         wf(l+p) 


(71) 


Upon  substituting  (71)  in  (70)  and  (69)  and  comparing  with  (56),  the  relation 
between  C8  and  C  is  found  to  be 


376  PERIODIC    ORBITS. 

It  is  seen  from  (56)  that  when  C  is  expanded  as  a  power  series  in  m,  or 
in  m1/3  if  a/ A  is  eliminated  by  the  last  equation  of  (6),  it  starts  off  with  a 
term  which  is  independent  of  m.  Therefore  C0,  the  Jacobian  constant  for 
the  integral  in  the  ordinary  form,  is  expansible  as  a  power  series  in  m1/3 
and  is  infinite  for  m  =  0.  The  three  periodic  orbits,  of  which  two  are 
complex  for  \m\  sufficiently  small,  corresponding  to  the  three  determinations 
of  a/A  in  (6),  coincide  and  branch  at  m  =  0  or  C0  =  oo .  Since  the  coordinates 
in  the  periodic  orbits  are  analytic  functions  of  w1/3,  and  w1/3  is  an  analytic 
function  of  C0  through  the  inversion  of  (72),  it  follows  that  the  coordinates 
in  the  periodic  orbits  are  analytic  functions  of  Ca.  One  branch-point  is  at 
C0  =  oo .  In  the  special  problem  treated  by  Darwin,*  in  which  the  ratio  of 
the  finite  masses  is  10  to  1,  he  found  by  computation  in  the  case  of  the 
orbits  around  the  smaller  finite  mass  that  there  is  another  branch-point 
for  a  certain  value  of  C0,  at  which  the  complex  orbits  first  become  real  and 
coincident,  and  then  real  and  distinct. 

190.  Applications  to  the  Lunar  Theory. — In  the  development  of  the 
Lunar  Theory  the  differential  equations  have  been  so  treated  that  the 
resulting  expression  for  the  distance,  or  its  reciprocal,  is  a  sum  of  terms  which 
involve  the  time  only  under  the  cosine  and  sine  functions.  The  longitude 
involves  terms  of  the  same  type  and  the  time  multiplied  by  a  constant  factor. 
Considering  the  problem  in  the  plane  of  the  ecliptic,  there  are  terms  whose 
period  is  equal  to  one-half  the  synodic  period  of  the  moon.  They  are  known 
as  the  variational  terms.  Now  the  period  of  the  periodic  orbits  which  have 
been  found  above  is  the  synodic  period  of  the  revolution  of  the  infinitesimal 
body,  or  twice  that  of  the  variational  terms.  The  terms  of  the  solutions 
which  are  of  even  degree  in  n  have  the  period  of  half  the  synodical  period  of 
revolution.  The  variational  terms  in  fact  belong  to  the  class  of  periodic 
orbits  treated  here.  The  detailed  comparison,  up  to  m9,  with  the  work  of 
Delaunay  was  made  in  the  Transactions  of  the  American  Mathematical 
Society,  vol.  VII  (1906),  p.  562,  and  perfect  agreement  was  found  except  in 
the  coefficients  of  the  higher  powers  of  m,  where  errors  are  almost  unavoid- 
able in  Delaunay's  complicated  method. 

Hill  wrote  a  remarkable  series  of  papers  on  the  Lunar  Theory  in  the 
American  Journal  of  Mathematics,  vol.  I  (1878),  in  which  he  proposed  to 
start  from  the  variational  orbit,  instead  of  from  an  ellipse,  as  an  intermediate 
orbit  for  the  determination  of  the  motion  of  the  moon.  The  elliptic  orbit 
as  an  intermediate  orbit  came  down  from  Newton  and  his  successors,  and 
the  inertia  of  the  human  mind  is  such  that  it  was  retained  for  over  a  century 
in  spite  of  the  fact  that  it  has  little  to  recommend  it.  Hill  has  the  great 
honor  of  initiating  a  new  movement  which,  it  seems  certain,  will  be  of  the 
highest  importance. 

•Ada  Mathematica,  vol.  XXI  (1897),  pp.  99-242. 


1M  I\III>I\IAL   SATELLITES   AND    INKKItloK   PLANETS.  877 

The  results  obtained  by  Hill  are  coextensive  with  those  given  here  if 
\ve  put  M  =  0  and  77  =  1  in  the  latter  series.  The  method  employed  by  Hill 
was  entirely  different  from  that  of  this  chapter.  It  was  convenient  in  prac- 
tice, but  its  validity  can  not  easily  be  established.  The  same  method  was 
extended  by  Broun  to  include  terms  which  contain  M  as  a  factor*  to  the 
first,  second,  and  third  degrees.  A  comparison  of  the  results  obtained  by 
the  met  hods  of  this  paper  with  those  of  several  writers  on  the  Lunar  Theory, 
especially  in  the  coefficient  of  a/ A  which  converges  most  slowly,  will  be 
found  in  the  Transactions  of  the  American  Mathematical  Society,  vol.  VII 
(1906),  p.  569. 

191.  Applications  to  Darwin's  Periodic  Orbits. — In  Darwin's  compu- 
tations, f  the  ratio  of  the  masses  of  the  finite  bodies  was  ten  to  one.  It  is 
found  from  the  definition  of  »j  and  the  last  equation  of  (6)  that  for  the 
motion  around  the  smaller  of  the  finite  bodies 

a_  _  /    ml    \l/l  /WV*...  fjA1*  (   m    y 
A      Vmi+Tnt/     \n/        \U/     \i+m/ 

Darwin  defined  his  orbits  by  the  value  of  the  Jacobian  constant,  and 
their  periods  were  found  from  the  detailed  computations.  In  comparing 
with  his  work  it  is  simpler  to  take  the  periods  which  he  obtained  and  to  find 
the  orbits  from  equation  (43).  The  comparison  will  be  made  first  with  his 
"Satellite  .A"  for  the  Jacobian  constant  in  his  notation  equal  to  40.5,  loc.  cit., 
p.  199.  The  synodic  period  was  found  to  be  61°  23'=  61.383°,  where  the 
period  of  the  finite  bodies  is  360°.  Therefore 

ro-  «yp  =0.17051,          f  =  (^"(^ST)" -0-12449.  (74) 


The  m  for  this  orbit  is  more  than  twice  that  occurring  in  the  Lunar  Theory. 
With  these  values  of  the  constants  substituted  in  the  series  of  §186,  it  is 
found  that 

r  =  0.1 2427 +0.00652  cosr-0.00420  cos2T+0.00004  cos3r 

-0.00006  cos  4r+ 

(75) 
w=  -0.12062  sin  T+ 0.05079  sin2r-0.00184  sin3r 

+  0.00095  sin  4r+  •  •  • 

The  infinitesimal  body  is  in  a  line  with  the  finite  bodies  and  between 
them  when  r  =  0.  The  value  of  r  at  this  time  is  found  from  (75)  to  be 
r(0)  =0.12657.  The  corresponding  value  given  by  Darwin  is  0.1265.  The 
infinitesimal  body  is  in  opposition  at  r  =  ir,  and  it  is  found  from  (75)  that 
r(*0  =  0.11345.  Darwin's  value  is  0.1135.  These  agreements  show  that 
the  "Satellites  A"  are  of  the  class  treated  in  this  chapter. 

•American  Journal  of  MaUiemaKct,  vol.  XIV  (1891),  pp.  14O-160. 
\.\cta  MaUumatica,  vol.  XXI  (1887),  pp.  99-242. 


378  PERIODIC   ORBITS. 

In  a  retrograde  orbit  having  the  same  sidereal  period,  the  expression 
for  m  is 

-N        -N/n 

n+N      l+N/n 
In  this  case 

N  613.83 


n        61.383  +  360' 

therefore 

m=  -0.12715. 

The  value  of  a/  A  is  the  same  as  before,  and  the  series  for  r  gives 

r  =  0.12412  -0.00057  cos  r-  0.00158  cos  2r-  0.00000  cos  3r 

-0.00001  cos  4r+  •  • 

(77) 
w  =  0.00820  sin  r+0.01654  sin  2r+0.00047  sin  3r 

+0.00010  sin  4r+ 

which  are  seen  to  converge  somewhat  more  rapidly  than  the  series  of  (95). 
No  retrograde  orbits  were  computed  by  Darwin  in  his  memoir  in  the  Ada 
Mathematica. 

Comparison  will  also  be  made  with  one  of  Darwin's  "Planets  A."     In 
this  case 

mt  =  10,        m2=l.         A  =  l, 


The  orbit  will  be  taken  for  which  the  Jacobian  constant  is  40.0.     The  period 
given  by  Darwin  (loc.  cit.  p.  225)  is  154°  13'.     Therefore 


m  =  =  0.42838,        -    =  0.43404.  (78) 

ouU  xL 

With  these  values  of  the  parameters,  the  series  for  r  gives 

r  =  0.43373+0.00776  COST-  0.01286  cos2r-  0.00104  cos  3r+  ----        (79) 
From  this  series  it  is  found  that 

r(0)  =  0.42759,        r(ir)  =  0.41415. 

Darwin's  results  in  the  respective  cases  were  r(0)  =  0.423  and  r(r)=  0.4140. 
The  agreement  of  these  results  shows  the  identity  of  his  "Planets  A" 
and  the  orbits  covered  by  the  analysis  of  this  chapter. 


CHAPTER  XII. 

PERIODIC  ORBITS  OF  SUPERIOR  PLANETS. 

192.  Introduction.    The  preceding  chapter  was  devoted  to  the  consid- 
ation  of  the  motion  of  an  infinitesimal  body  subject  to  the  attraction  of  two 
Unite  bodies  which  revolve  in  circles.    The  periodic  orbits  whose  existence 
\\as  there  proved  inclose  only  one  of  the  finite  bodies,  and  they  are  more 
nearly  circular  the  smaller  their  dimensions  and  the  shorter  their  periods. 

The  present  chapter  also  will  be  devoted  to  the  consideration  of  the 
motion  of  an  infinitesimal  body  subject  to  the  attraction  of  two  finite  bodies 
which  revolve  in  circles;  but  the  periodic  orbits  now  under  discussion  inclose 
lioth  of  the  finite  bodies  and  are  more  nearly  circular  the  larger  their  dimensions 
and  the  longer  their  periods.  There  are  three  families  of  orbits  of  this  class 
in  which  the  motion  is  direct,  and  three  in  which  it  is  retrograde.  For  small 
values  of  the  parameter  in  terms  of  which  the  solutions  are  developed,  only 
one  family  each  of  the  direct  and  of  the  retrograde  orbits  is  real. 

The  mode  of  treatment  of  the  problem  of  this  chapter  is  similar  to 
that  of  the  preceding.  A  certain  parameter  n  naturally  enters  the  problem. 
When  n  is  zero,  the  problem  reduces  to  that  of  two  bodies,  which  admits  a 
circular  orbit  as  a  periodic  solution.  The  existence  of  the  analytic  continu- 
ation of  this  orbit  with  respect  to  the  parameter  n  is  proved,  and  direct 
methods  of  constructing  the  solutions  are  developed.  It  is  shown  also  how 
th«-  integral  can  be  used  as  a  check  on  the  computations,  or  as  a  substitute 
for  one  of  the  differential  equations  in  the  construction  of  the  solutions. 

The  results  of  the  preceding  chapter  were  directly  applicable  to  the 
Lunar  Theory;  those  of  this  chapter  have  no  direct  bearing  on  the  practical 
problems  of  the  solar  system,  at  least  as  they  are  at  present  treated.  Their 
chief  value  at  present  is  that  they  cover  a  part  of  the  field  of  the  problem 
of  three  bodies  in  which  one  is  infinitesimal  and  in  which  the  finite  bodies 
revolve  in  circles. 

193.  The  Differential  Equations. — Let  the  origin  of  coordinates  be  at 
the  center  of  gravity  of  the  finite  bodies  w,  and  m, ,  and  take  the  xy-plane 
as  the  plane  of  their  motion.     Suppose  the  infinitesimal  body  moves  in  the 
///-plane.     Let  the  coordinates  of  w,,  w,,  and  the  infinitesimal  body  be 
(xi,  Vi),  fe>  l/i),  and  (x>  I/)  respectively.    Then  the  differential  equations  of 
motion  for  the  infinitesimal  body  are 


(1) 


_dU  -.  Tr_i  . 

de"~"dx"          d?  "~  dy'  r, 


r,  =  V(x-xt)t+(y-Vi)t, 

379 


380  PERIODIC    ORBITS. 

L^        r=V^W,  R^V^W^^r^R, 


-y*Y,       Rt=Vxl+yl  = 

Then,  in  polar  coordinates,  equations  (1)  become 


(2) 


_          =  -_  /o\ 

df  "     V         "  dr  '         r  df~*     dt  dt  ~  r  dv  ' 

The  potential  function  U  will  now  be  developed.     From  (1)  and  (2)  it 
is  found  that 


ff  {i[1+3cos2(0_«1)] 


(4) 


Then  equations  (3)  become 


_  r  +         t  =  _  ,  l  +  3  cos  2  (,-,, 

r  LL4 


cos    »- 


(5) 
d2?; 


-ji2j7jT  -T 

df        dtdt          mi+rrh  r4  [ 


If  the  orbits  of  ml  and  m2  are  circles,  which  is  assumed  to  be  the  case, 
equations  (1)  admit  the  Jacobian  integral 

dy_  ,. 
dt      ydt 


It  follows  that  r^  is  the  mean  motion  of  the  finite  bodies  and  that  vt  =  n1(t  —  0 . 
In  polar  coordinates  the  integral  becomes 

r'2+/V2— 2n,r*v'  =  2U— C,  (7) 

1  7  \      / 

where  the  primes  indicate  derivatives  with  respect  to  t. 

When  the  right  members  of  (5)  are  put  equal  to  zero,  the  equations 
admit  the  particular  solution          

where  n  is  the  angular  velocity  of  the  infinitesimal  body  in  its  orbit  and  t0  is  an 
arbitrary  constant.  It  will  be  supposed  that  n  is  given  by  the  observations, 
or  that  its  value  is  assumed,  and  that  a  is  determined  by  the  second  equation 
of  (8).  The  constant  a  has  three  values,  only  one  of  which  is  real. 


PKIUODIC  OKHI!.-   «>1     >l   I'KKIOR    PLANETS.  381 

New  variables,  p,  9,  and  T,  and  new  constants,  n  and  M  ,  will  be  introduced 
by  the  equations 


Af.     (9) 

I  it 

It  follows  from  (6),  (8),  and  (9)  that 


and  equations  (5)  become 


";  (io) 


(ii) 


where  the  dots  over  the  letters  indicate  derivatives  with  respect  to  T.  These 
equations  are  valid  for  the  determination  of  the  motion  of  the  infinitesimal 
body  provided  |M|<!-  The  right  members  of  equations  (11)  involve  only 
cosines  and  sines  respectively  of  integral  multiples  of  r+6.  The  parts  in  the 
brackets  proceed  according  to  powers  of  /i17',  the  coefficients  of  even  powers 
of  M1'3  in  the  first  and  second  equations  being  cosines  and  sines  respectively 
of  even  multiples  of  T,  and  the  coefficients  of  odd  powers  of  MV>  being  cosines 
and  sines  respectively  of  odd  multiples  of  T. 

194.  Proof  of  the  Existence  of  Periodic  Solutions.  —  Suppose  p  =  /3,  p  =  0, 
0  =  0,  0  =  7  at  r  =  0,  and  let  the  solution  of  (11)  be  written  in  the  form 


P=/03,T;T),         6  =  ^(0,  y;  T).  (12) 

Now  make  the  transformation 

P  =  Pl,        0=-0lt        r=-rl.  (13) 

The  resulting  equations  have  precisely  the  form  (11).  Consequently  their 
solutions  with  the  initial  conditions  pi  =  /3,  p,  =  0,  6  =  0,  0,  =  7  are 

Pi=/(0,  7;T,)=/G3,  T;-T)  =  P,         6l  =  <p(0,T,Tt)=<f>(f3,y;-T)  =  -0.   (14) 

Therefore,  with  these  initial  conditions,  p  is  an  even  function  of  T,  and  6  is 
an  odd  function  of  T.  The  orbit  is  symmetrical  with  respect  to  the  p-axis 
both  geometrically  and  in  r.  Such  an  orbit  will  be  called  symmetrical, 
whether  it  is  periodic  or  not. 


382  PERIODIC    ORBITS. 

Now  consider  the  conditions  for  a  closed  symmetrical  orbit.  Since  the 
right  members  of  (11)  involve  only  sines  and  cosines  of  integral  multiples 
of  r,  sufficient  conditions  that  in  symmetrical  orbits  p  and  6  shall  be  periodic 
with  the  period  2jir  are 

p  =  /03,  7  ;»=(),         0  =  *>G8,7;./V)  =  0;  (15) 

and  these  conditions  are  necessary,  provided  they  are  distinct. 

In  order  to  examine  the  solutions  of  (15),  it  is  convenient  to  use  par- 
ameters other  than  /3  and  7.  Suppose  that,  at  r  =  0, 

r  =  a(l+p)=a(l-fa)(l  —  e),          f  =  ap  =  0, 


M     ,  A_    M  Vi+e 

~T(/  —  , 
1-M  I-/ 


(16) 

It  follows  that  a(l+a)  and  e  are  the  major  semi-axis  and  eccentricity  of  the 
elliptic  orbit  which  would  be  obtained  if  the  right  members  of  equations 
(11)  were  zero.  Because  of  the  well-known  properties  of  the  solutions  of 
the  two-body  problem  in  terms  of  these  elements,  the  properties  of  the 
general  solutions,  so  far  as  they  do  not  depend  upon  the  right  members  of 
(11),  are  known.  These  properties  will  be  important  in  solving  the  conditions 
for  periodic  solutions. 

Equations  (11)  are  regular  in  the  vicinity  of  M  =  0,  p  =  0,  p  =  0,  6  =  0, 
6  =  0  for  all  values  of  T.  It  follows  that  the  moduli  of  a,  e,  and  /x1/3  can  be 
taken  so  small  that  the  solutions  will  be  regular  while  T  runs  through  any 
finite  preassigned  range  of  values.  We  shall  choose  as  the  interval  for  T  the 
range  Q^.T^2jir  and  integrate  (11)  as  power  series  in  o,  e,  and  MI/S,  vanishing 
with  a  =  e  =  /j.l/3  =  0.  That  is,  the  results  will  be  the  analytic  continuation 
with  respect  to  these  parameters  of  the  particular  solution  r  =  a,  v  =  nt,  which 
exists  when  ju,  =  0.  The  results  may  be  written  in  the  form 

>  =  p3(a,e,MV3;r),  ) 


where  pl}  .  .  .  ,  p^  are  power  series  in  a,  e,  and  M1/3>  with  T  in  the  coefficients. 
The  conditions  for  a  periodic  solution,  (15),  become 

p3  (a,  e,  M1/3J »  =  0,         p3  (a,  e,  M1/3;»  =  0.  (18) 

It  will  be  shown  that  these  equations  can  be  solved  for  a  and  e  as  power 
series  in  /zI/3,  vanishing  with  ///3  =  0,  which  converge  if  the  modulus  of  /*l/3  is 
sufficiently  small. 

Since  the  right  members  of  (11)  carry  /i10/3  as  a  factor,  the  part  of  the 
solution  depending  on  the  right  members  will  be  divisible  by  M10/3-  If 
the  right  members  of  (11)  were  zero  and  if  the  solution  were  formed  with 
the  initial  conditions  (16),  the  mean  angular  motion  of  the  infinitesimal 
body  in  its  orbit  would],be 

(19) 


PKUIODIC   (IHBIT8  OF  SUPERIOR    PLANETS.  383 

Consequent  ly,  from  the  solution  of  the  two-body  problem,  it  follows  that 
equations  (18)  have  the  form 


=  0, 

(20) 


when-  the  unwritten  parts  in  the  brackets  are  sines  of  multiples  of  vjr,  and 
carry  e1  as  a  factor. 

Upon  referring  to  (19),  it  is  observed  that  the  first  equation  of  (20)  is 
divisible  by  M'»  and  the  second  by  n.  After  dividing  by  these  factors  the 
equations  are  still  satisfied  by  a  =  e  =  n  =  0;  moreover,  the  determinant  of 
their  linear  terms  in  a  and  e  is 


0  ,      jw 

3V      9^  A 

—  — 7H 

«w      J      J 


(21) 


Therefore,  besides  the  solution  n  =  0,  equations  (20)  have  a  unique  solution 
for  a  and  e  as  power  series  in  /x!/*,  vanishing  with  /nVl  =  0,  which  converge  for 
the  modulus  of  MV>  sufficiently  small.  These  power  series  carry  nv*  as  a 
factor,  and  can  be  written  in  the  form 

a  =  M4*P,(M*),  e-|i*W*).  (22) 

Upon  substituting  these  series  in  the  right  members  of  (17),  which  vanish 
with  a  =  e  =  nt/t  =  Q,  the  result  is 

P  =  MV,QI(MV,;  T)|          8- MM*',  T).  (23) 

The  series  Q,  and  Qt  are  periodic  in  r  with  the  period  2jir  because  the  con- 
ditions that  the  solutions  shall  have  this  period  have  been  satisfied.  Since 
(17)  converge  for  all  O^T^2jr  if  the  moduli  of  a,  e,  and  M'*  are  sufficiently 
small,  and  since  the  expressions  for  o  and  e  given  in  (22)  vanish  for  M  =  0,  it 
follows  that  the  modulus  of  n:/t  can  be  taken  so  small  that  the  series  (23)  con- 
verge for  all  T  in  the  interval;  and  since  they  are  periodic  with  the  period  2jr, 
the  convergence  holds  for  all  finite  values  of  r. 

The  integer  j  has  so  far  been  undetermined.  When  j  is  unity,  the 
periodic  solutions  exist  uniquely  and  their  period  is  2*-.  When .;'  is  greater 
than  unity  the  periodic  solutions  also  exist  uniquely.  Since  the  periodic 
orbits  for  j  greater  than  unity  include  those  for  j  equal  to  unity,  and  since  in 
both  cases  there  is  precisely  one  periodic  orbit  for  a  given  value  of  ^, 
it  follows  that  all  the  symmetrical  periodic  orbits  of  the  class  under  consideration 
have  the  period  2*  in  the  independent  variable  T. 


384  PERIODIC   ORBITS. 

It  follows  from  (6)  and  (9)  that  T+6  =  v—v1.  Since  in  the  periodic 
solution  6  is  periodic  with  the  period  2w,  the  period  of  the  solution  is  the 
synodic  period  of  the  three  bodies.  Hence,  if  the  motion  of  the  infinitesimal 
body  is  referred  to  a  set  of  axes  having  their  origin  at  the  center  of  gravity 
of  the  system  and  rotating  in  the  direction  of  motion  of  the  finite  bodies  at 
the  angular  rate  at  which  they  move,  and  if  the  :r-axis  passes  through  the 
finite  bodies,  then  the  periodic  orbit  of  the  infinitesimal  body,  which  has 
been  proved  to  exist,  will  be  symmetrical  with  respect  to  the  z-axis.  Since, 
by  hypothesis,  a  >  R,  it  follows  from  (6)  and  (8)  that  n:>n.  Therefore,  even 
if  the  motion  of  the  infinitesimal  body  is  forward  with  respect  to  fixed  axes, 
it  is  retrograde  with  respect  to  the  rotating  axes. 

It  is  supposed  that  the  period  of  the  finite  bodies,  and  therefore  nl}  is 
given  in  advance  and  remains  fixed.  The  variation  of  the  parameter  p" 
corresponds  to  a  variation  of  the  period  of  the  infinitesimal  body  defined 
by  n.  If  the  motion  with  respect  to  fixed  axes  is  forward,  n  has  the  same 
sign  as  nl}  and  nl/3  has  three  values,  one  of  which  is  real  and  positive  while 
the  other  two  are  complex.  If  the  motion  is  retrograde,  nvs  has  three 
different  values,  one  of  which  is  real  and  negative  while  the  other  two  are 
complex.  Therefore,  for  a  given  period,  there  are  six  symmetrical  orbits, 
three  direct  and  three  retrograde;  and  for  small  /x  one  direct  orbit  is  real  and 
one  retrograde  orbit  is  real,  while  in  the  others  the  coordinates  are  complex. 
This  means,  of  course,  that  the  corresponding  solutions  do  not  exist  in  the 
physical  problem.  The  coordinates  of  the  complex  orbits  are  conjugate  in 
pairs.  For  a  certain  value  of  /z1/3  they  may  become  equal,  and  therefore 
real,  and,  for  larger  values  of  //3,  real  and  distinct. 

Upon  transforming  the  integral  (7)  by  (9),  it  is  found  that 


'*  111"  _      .  .    ~l         .          1/1,0.   —    ill,.  It.       "  I      _  *  .          _*.  if-      .  \ 

(24) 


+5cos3(r+0)]+  •••]-<?„ 


where  (7  =  n\(\  —  /z^a2^.  It  follows  from  this  equation  and  (23)  that  Cl 
can  be  expanded  as  a  power  series  in  /j.l/3,  vanishing  with  nl/3.  The  term  of 
lowest  degree  in  MVS,  after  substituting  (23),  is  ju3/3-  Therefore,  MI/S  can  be 
expanded  as  a  power  series  in  C1/3.  For  C'1  =  0,  the  three  branches  of  the 
function  are  the  same.  Since  a  =  R/fj?/3,  the  relation  between  C  and  Cl  is 


C  =  r=          =          [l  +power  series  in  M].  (25) 

From  this  it  follows  that  C  =  <x>  for  /i1/3  =  0.     Therefore  the  periodic  orbits 
branch  at  C=  oo  ,  and  there  are  two  cycles  of  three  each. 


PERIODIC   OKBITS   OF   Kri'KHluK    PLANETS.  .'>S"> 

195.  Practical  Construction  of  the  Periodic  Solutions.  It  has  been 
proved  that  the  symmetrical  periodic  solutions  under  discussion  are  expres- 
sible in  the  form 

(26) 


whore  the  p,  and  8,  arc  functions  of  r.  Since  these  series  are  periodic  and 
converge  for  all  IM'*]  sufficiently  small,  it  follows  that  each  p,  and  0,  separately 
is  periodic;  that  is, 

p,(r+2T)==p,(T),         6l(T+2ir)=6l(r).  (27) 

In  every  closed  orbit  there  are  points  at  which  dp/dr  =  0.     Suppose  / 
of  (9)  is  so  determined  that  this  condition  is  satisfied  at  T  =  0;  it  will  follow 
from  this  and  the  convergence  of  (26)  for  all  |M'/J|  sufficiently  small  that 

P,  =  0  (t-4,  ...  «).         (28) 

In  the  symmetrical  periodic  orbits  the  value  of  8  is  zero  at  r  =  0.  But 
this  condition  will  not  be  imposed,  because  the  general  periodic  orbits,  whose 
initial  conditions  are  not  specialized,  include  those  which  are  symmetrical; 
and  in  the  construction  it  will  appear  that  the  conditions  for  symmetry  are 
a  consequence  of  those  for  periodicity.  Hence  all  the  periodic  orbits  of  the 
class  under  consideration  are  symmetrical. 

Equations  (26)  are  to  be  substituted  in  (11),  arranged  as  power  series 
in  M'/J,  and  the  coefficients  of  the  several  powers  of  MI/J  set  equal  to  zero. 
The  coefficients  of  MV|  set  equal  to  zero  give  the  equations 

p>0,        0>0.  (29) 

The  solutions  of  these  equations  satisfying  (27)  and  (28)  are 

P«  =  a«,        04  =  &4,  (30) 

where  a,  and  &«  are  so  far  undetermined  constants. 

The  coefficients  of  /iw,  .  .  .  ,  /**/l  are  the  same  as  (29)  except  for  their 
subscripts,  and  their  solutions  satisfying  (27)  and  (28)  are  similarly 

Pj  =  aj,        e,  =  b,  a-5,  .  .  .  ,  9),        (31) 

where  all  the  a,  and  b,  are  so  far  undetermined  constants. 
The  coefficients  of  M'O/I  give  the  equations 

pw  =  3a4-fA/[l+3cos2r],        0,.  =  -  f  M  sin  2r.  (32) 

In  order  that  the  solution  of  the  first  of  these  equations  shall  be  periodic 
the  condition 

a4=±M  (33) 

must  be  imposed,  which  uniquely  determines  the  constant  o4.  Then  the 
solution  of  (32)  satisfying  (27)  and  (28)  is 

P,,=Oi.+  ^cos2T,        010-&lo+fMsin2r,  (34) 

where  a,0  and  &„  are  as  yet  undetermined. 


386  PERIODIC    ORBITS. 

The  coefficients  of  /x11/3  are  defined  by  the  equations 

pn  =  3a6,         0ii  =  0, 
from  which  it  follows  that 

o6  =  0,        Pn  =  an,         0ii  =  &n,  (35) 

where  au  and  6n  are  as  yet  undetermined. 

The  coefficients  of  MI2/3  in  the  solutions  satisfy  the  equations 


where  a12  and  612  remain  so  far  undetermined. 

In  a  similar  way  it  is  found  from  the  coefficients  of  /x13/3  that 


Upon  imposing  conditions  (27)  and  integrating,  it  is  found  that  a6  =  0,  and 

(36) 
L   si 

ined. 

j  coefficients  of  fj.n/3  that 

+  ^Msin2r,  (37) 

where  a13  and  bl3  remain  undetermined  at  this  step. 

So  far  all  the  b}  have  remained  arbitrary,  and  it  is  necessary  to  carry  the 
integration  one  step  further  in  order  to  see  how  they  are  determined.  The 
coefficients  of  MH/3  are  defined  by 

(38) 
0H=  —  a4010  —  3M  [64cos2r  —  2a4sin2r]. 

Upon  substituting  the  values  of  a4  and  010  from  (33)  and  (34),  imposing  the 
conditions  (27)  and  integrating,  it  is  found  that  a8=  —  ^M2,  and 


(39) 

The  condition  (28)  for,;' =  14  gives  the  equations 

&4=  0,        PU  =  OH  —  ^If2cos2r,         014  =  614—  ^M2sin2T.  (40) 

ID  Ow 

It  is  found  in  a  similar  way  from  the  coefficients  of  /xu/3  that  &6  =  0,  and 


(41) 


where  a15  and  bK  remain  so  far  arbitrary. 


c.KHITS    ()K    ST  I'M:  IOR    PLANETS. 

It  will  l>e  observed  that,  so  far  as  the  computation  has  l>een  carried,  the 
coefficients  of  the  p.  are  cosines  of  integral  multiples  of  r,  and  that  the  coeffi- 
cients of  the  8j  ,  except  for  the  undetermined  additive  constants,  are  sines  of 
integral  multiples  of  T.  In  the  computation  of  p,  the  periodicity  conditions 
have  uniquely  determined  a/,,,  and  the  condition  p,  =  0  at  T  =  0  has 
required  that  fe,_,0  =  0.  It  will  now  be  shown  that  these  properties  are 
general.  Suppose  p4  ,  .  .  .  ,  p.;  04,  .  .  .  ,  0.  have  been  computed  and  that 
the  coefficients  are  all  known  except  a,-!,  .  .  .  ,  a,,  which  enter  additively 
in  p,_,,  .  .  .  ,  p.  respectively,  and  6._,,  .  .  .  ,  bm,  which  enter  additively  in 
0,_,,  .  .  .,  0.  respectively.  The  differential  equations  for  the  determination 
of  p..,.,  and  0.+,  are 

p.+l  =  3a._,+  fM&._,sin2T+F.+1(T),        0.+l  =G,+1  (r),  (42) 

where  /•'„+,  (T)  and  Gn+l(r)  are  entirely  known  functions  of  T.  It  follows  from 
the  assumptions  respecting  p4,  .  .  .  ,  p.;  04,  .  .  .  ,  0.  and  the  properties  of 
equations  (11)  that  Fu+l(r)  is  a  sum  of  cosines  of  integral  multiples  of  T,  and 
that  Gn+i(r)  is  a  sum  of  sines  of  integral  multiples  of  T.  Hence  they  may  be 
written  in  the  form 


FH.,(T)  =  2  A**"  COSJr,          (?.+,(T)  =  2  B^  Bin  jr. 

In  order  that  the  solution  of  the  first  equation  of  (42)  shall  be  periodic 

the  condition 

=  0  (43) 


must  be  imposed,  and  this  condition  uniquely  determines  a,_,. 

After  equation  (43)  has  been  satisfied,  the  solution  of  the  first  equation 
of  (42)  is 

I  =  o.+l-    M6._,8in2r+2  a^"  cos  jr.          <+''=  -U^".      (44) 


The  condition  p  =  0  at  T  =  0  makes  it  necessary  to  take 

6.-,  =  0.  (45) 

Then  p.+i  is  completely  determined  except  for  the  additive  constant  0.+,  , 
and  it  is  a  sum  of  cosines  of  integral  multiples  of  T. 
The  solution  of  the  second  equation  of  (42)  is 

0.+1  =  6.+1+20r+1)«n;V,        ff*n  -  -^T11.  (46) 

Hence  0,+,  is  a  sum  of  sines  of  integral  multiples  of  T,  except  for  the  unde- 
termined constant  6.+1  ,  which  must  be  put  equal  to  zero  in  order  to  satisfy 
the  condition  on  p.+II.  These  results  lead,  by  induction,  to  the  conclusion 
that  the  p,  and  9,  0'=4,  .  .  .  GO)  are  sums  of  cosines  and  sines  respectively 
of  integral  multiples  of  T  whose  coefficients  are  uniquely  determined. 


388  PERIODIC   ORBITS. 

From  the  properties  of  the  solutions  which  have  just  been  established, 
it  follows  that  not  only  is  p  =  0  at  T  =  0,  but  also  0(0)  =  0.  Therefore  these 
periodic  orbits  are  the  symmetrical  orbits  whose  existence  was  established 
in  §194.  In  the  construction  it  was  not  assumed  that  the  orbits  were 
symmetrical,  and  since  this  property  is  a  necessary  consequence  of  the 
periodicity  conditions,  it  follows  that  all  periodic  solutions  which  are  expan- 
sible as  power  series  in  ju1/3  are  symmetrical.  It  is  easily  shown,  by  direct 
consideration  of  the  construction  of  periodic  solutions,  that  they  can  not  be 
expanded  as  power  series  in  /*1/3  except  when  j  is  a  multiple  of  3,  and  that  then 
they  reduce  to  those  found  above. 

196.  Application  of  the  Integral.  —  The  differential  equations  admit  the 
integral  (24),  which,  for  brevity,  can  be  written  in  the  form 


It  follows  from  the  form  of  (24)  and  the  expansions  (26)  that  the  left  member 
of  this  equation  can  be  developed  as  a  power  series  in  ju1/3,  giving 


»+  •  •  •   =0.  (47) 

Since  the  p}  and  0,  are  sums  of  cosines  and  sines  respectively  of  integral 
multiples  of  T,  and  since  p  enters  in  (24)  only  in  the  second  degree  and  6  only 
in  even  degrees,  it  follows  that  the  Fj  are  sums  of  cosines  of  integral  multiples 
of  T.  Equation  (47)  is  an  identity  in  ;u1/3,  whence 

Fn=SCffcos./T=0  (n=0,  ...  oo). 

Since  these  equations  hold  for  all  values  of  T,  it  follows  that 

Cf  =  0  (n  =  0,  ...   «.  ;  j  =  0,  .  .  .   oo).  (48) 

The  C*°  are  functions  of  the  <  ,  .  .  .  ,  af  and  #0),  .  .  .  ,  p™  .  Hence 
equations  (48)  can  be  used  as  check  formulas  on  the  computation  of  the 
coefficients  of  the  solutions. 

Equations  (48)  can  be  used  in  place  of  the  second  equation  of  (11)  for 
the  determination  of  the  $B),  the  coefficients  of  the  trigonometric  terms  in 
the  expression  for  0n  .  Suppose  p4  ,  .  .  .  ,  pn-t  and  04  ,  .  .  .  ,  0n_1  have  been 
determined  except  for  additive  constants  in  pB_6  ,  .  .  .  ,  pB_!  .  It  follows  from 
(24)  that  Fn  is 

F.  =  -  26n  +  Pa  (P,,  PI,  e},  0,)  (j=4,,  .  .  .  ,  n-  l), 

where  Pn  is  a  polynomial  in  the  arguments  indicated.  Consequently 
equations  (48)  are  of  the  form 


which  uniquely  determine  the  $"'. 


CHAPTER  XIII. 

A  CLASS  OF  PERIODIC  ORBITS  OF  A  PARTICLE  SUB- 
JECT TO  THE  ATTRACTION  OF  ;/  SPHERES 
HAVING  PRESCRIBED  MOTION. 


BY  WILLIAM  RAYMOND  LONGLEY. 


197.  Introduction. — The  restricted  problem  of  three  bodies  furnishes 
naturally  the  starting-point*  for  the  consideration  of  the  periodic  orbits 
of  an  infinitesimal  body,  or  particle,  which  is  subject  to  the  Newtonian 
attraction  of  certain  finite  spheres  whose  motion  is  supposed  to  be  known. 
The  two  finite  bodies  are  supposed  to  revolve  in  circles  about  their  common 
center  of  mass,  and  the  motion  of  the  particle  is  restricted  to  the  plane  in 
which  the  finite  bodies  move.  One  class  of  orbits  occurring  in  this  problem 
is  that  in  which  the  particle  revolves  about  one  of  the  finite  bodies,  and  for 
the  consideration  of  these  orbits  it  is  convenient  to  refer  the  motion  of  the 
particle  to  a  plane  rotating  with  the  angular  velocity  of  the  finite  bodies. 
All  of  the  known  periodic  orbits  of  this  type  possess  one  and  only  one  line 
of  symmetry,  namely,  the  line  joining  the  finite  bodies,  and  this  property 
of  symmetry  plays  an  important  part  in  the  proof  of  their  existence  and  the 
construction  of  series  to  represent  them. 

The  purpose  of  this  chapter  is  to  generalize  the  restricted  problem  by 
introducing  into  the  plane  of  motion  more  than  two  finite  bodies.  The 
coordinates  of  the  finite  bodies  (spheres)  are  supposed  to  be  known  functions 
of  the  time,  that  is,  the  motion  of  the  spheres  is  prescribed.  For  the  analysis 
which  follows  the  nature  of  the  forces  producing  this  motion  is  unimportant. 
The  spheres  are  supposed  to  attract  the  particle  according  to  the  Newtonian 
law.  Besides  involving  additional  terms  in  the  disturbing  function,  this 
generalization  modifies  the  original  problem  by  introducing  cases  where 
the  periodic  orbits  have  no  line  of  symmetry,  and  cases  where  there  are 
more  lines  of  symmetry  than  one.  This  modification  necessitates  some 
changes  in  the  details  of  the  analysis  which  must  be  worked  out.  In  order 
to  avoid  cumbersome  notation,  the  analysis  will  be  developed  for  simple 
particular  cases  of  the  motion  of  spheres  under  their  Newtonian  attraction; 
with  slight  changes  it  is  applicable  to  more  general  types  of  prescribed 
motion  of  the  finite  bodies,  which  are  indicated  in  §207. 

•See  papers  by  Hill,  American  Journal  of  Malhematict,  vol.  1  (1878),  p.  245;  Darwin,  Ada  Mathcmatica, 
vol.  21  (1897),  p.  99;  and  Moulton,  Transaction*  of  the  American  Mathematical  Society,  vol.  7  (1906),  p.  537. 


390  PERIODIC    ORBITS. 

198.  Existence  of  Periodic  Orbits  Having  no  Line  of  Symmetry.  —  It 
was  shown  by  Lagrange*  that  an  equilateral  triangle  is  a  possible  configu- 
ration for  three  spheres  revolving  in  circles  about  their  common  center  of 
mass.  This  motion  of  three  finite  bodies  will  serve  to  illustrate  the  case 
when  the  periodic  orbits  of  the  particle  about  one  of  the  bodies  possess  no 
line  of  symmetry.  Let  the  masses  of  the  three  finite  bodies  moving  accord- 
ing to  the  equilateral-triangle  solution  be  denoted  by  M  ,  M1}  M,  ,  and  sup- 
pose the  particle  P  revolves  about  the  mass  M.  Suppose  also  that  the 
masses  Ml  and  M2  are  unequal.f 

With  reference  to  M  as  origin  and  an  axis  having  a  fixed  direction  in 
space,  let  the  polar  coordinates  of  Mlt  M2,  and  P  be  respectively  (Rlf  VJ, 
(Rt,  F2),  and  (r,  v).  The  coordinates  of  the  bodies  are  expressed  in  terms 
of  the  time,  t,  as  follows: 

R^R^A,        V^Vz-l=Nt,  (1) 

where 


Here  N  denotes  the  angular  velocity,  A  the  length  of  a  side  of  the  triangle, 
and  A;  is  a  constant  depending  upon  the  units  employed. 
The  differential  equations  of  motion  of  P  are 


df     T\dt)         r2        dr  '          1 'dt2  H      dt  dt  ==  r  dv  ' 
where 

_,2rMi    ,    Ms  _  MI,          /    _y\_Mt,         ,    _y\\ 

(3) 


Let  us  define  m  and  a  by  the  relations 

mv  =  N,          i/*a'  =  fcfM,  (4) 

where  v  is  a  quantity  to  be  assigned  later. 

By  the  substitution  v  =  w+V1  =  w  +  Nt  the  motion  is  referred  to  an 
axis  rotating  with  the  angular  velocity  of  the  finite  bodies  and  passing 
always  through  M^,  and  factors  depending  upon  the  units  employed  are 
eliminated  by  the  relations  r  =  ap,  vi  —  T.  On  making  these  substitutions  in 
equations  (2)  and  dividing  by  v"a,  the  differential  equations  of  relative 
motion  become 


*Prize  memoir,  Essai  sur  le  Probleme  des  Trois  Corps,  1772;  Coll.  Works,  vol.  6,  p.  229. 

flf  Mi  =M,  the  periodic  orbit  of  P  about  M  has  a  line  of  symmetry,  namely,  the  median  of  the  triangle 
from  the  vertex  M,  which  is  the  line  joining  M  to  the  center  of  mass  of  the  system.  For  the  treatment  of 
this  special  case  it  is  convenient  to  make  use  of  the  property  of  symmetry  and  to  employ  analysis  similar 
to  that  developed  in  §§202  and  203. 


PARTICLE   ATTRACTED    BY    «    SI'llKltKS.  391 

We  can  expand  <>  as  a  power  series  in  r/p/.l  which  is  convergent  for  all 
values  of  w  provided  the  distance  Ml'  =  np  is  l.-s-  than  .-1  ;  and  in  all  that 
follows  this  condition  is  supposed  to  he  satisfied.  The  expansion  has  the 

fiinu 


Let  X,  and  X,  he  defined  by  the  relations 

Af^X.M+A/,),        J/.-X.CAf.+Af,).  (6) 

From  equations  (1)  we  have 

Af.+Af.        . 


A* 

Then,  on  setting 

Ml  +  Mt 

- 


it  follows  that  the  second  members  of  equations  (5)  have  the  form 


(7) 
a1"  li  =  -/i:wlp[|xIsin2u;-|-|xl(-|p)|sinu>+5sin3u>)H 

!)}+•••] 


It  is  convenient  to  introduce  a  parameter  n  into  the  differential  equations 
(5)  by  the  relations 

m=n,        X,  =  XM,         J=IWi  (8) 

wherever  the  degree  of  a/A  is  higher  than  the  first.  The  quantities  X  and  ij 
are  numerical  constants.  By  relating  X,  and  /i  the  existence  proof  is  made 
to  depend  only  upon  general  properties  and  certain  terms  of  the  differential 
equations  which  involve  X,  ;  that  is,  upon  terms  in  the  disturbing  function 
which  are  due  to  the  body  M  ,  .  We  shall  consider  the  solution  of  equations 
(5)  as  power  series  in  the  parameter  jt.  The  differential  equations,  and 
consequently  also  the  solution,  do  not  represent  the  physical  problem  under 
consideration  for  any  value  of  the  parameter  except  the  one  satisfying  the 
relations  (8).  But  if  the  solution  is  valid  when  this  particular  numerical 
value  of  M  is  substituted,  then  it  is  a  solution  of  the  differential  equations 


392  PERIODIC   ORBITS. 

representing  the  physical  problem  and  therefore  has  a  physical  interpre- 
tation. The  generalization  of  the  parameter  a/A  is  merely  for  convenience 
in  having  finite  expressions  in  the  equations  which  determine  the  coefficients 
at  the  various  steps  in  the  solution. 

On  introducing  the  parameter  n  as  indicated,  equations  (5)  become 

d*p        fdw 
dr2        \  dr 
where 

v  F    fi  3  a  1  1 

L   I2  8  -4  J  J 

g--zA 


where  F  and  G  are  functions  of  the  indicated  arguments. 

Equations  (9)  are  periodic  in  w  with  the  period  2w  and  do  not  involve  T 
explicitly.  Suppose  that 

P  =  ^iO),          W  =  fa(r) 

is  a  solution.  Sufficient  conditions  that  the  solution  shall  be  periodic  with 
the  period  2pir  (where  p  is  an  integer)  are 

2T=tf(0),  j 

=#(0),  J 

where  \f/[  and  \]/'a  denote  derivatives  of  fa  and  \f/2  with  respect  to  T. 

When  n  —  0  a  periodic  solution,  which  will  be  called  the  undisturbed 
orbit,  is  known,  namely, 

P  =  l,        W  =  T,  (11) 

and  the  initial  conditions  are 

P  =  l,        p'  =  0,        w  =  Q,        w'  =  l.  (12) 

It  will  be  shown  by  the  process  of  analytic  continuation  that,  for  values  of 
fj.  different  from  zero,  but  sufficiently  small,  there  exists  a  periodic  solution 
which,  for  n  =  0,  reduces  to  equations  (11).  For  this  purpose  we  consider 
the  solution  of  equations  (9)  subject  to  the  initial  conditions 

P=l+]81,        p'  =  &,        w  =  &,         w'  =  l+pt,  (13) 

where  ft ,  j8j ,  ft ,  /34 ,  are  to  be  determined  as  functions  of  /*,  vanishing  with 
H,  so  that  the  conditions  of  periodicity  (10)  shall  hold.  It  follows  from  the 
differential  equations  that  the  solution  is  expressible  as  power  series  in  ft , 
ft  >  ft  >  ft  j  and  M  and  that,  for  sufficiently  small  values  of  the  parameters, 
the  series  are  convergent  for  all  values  of  T  from  0  to  2pir.  We  suppose 
that  this  condition  on  the  moduli  of  the  parameters  is  satisfied. 

For  the  determination  of  those  terms  in  the  series  which  involve  the 
initial  conditions  but  not  (j?,  it  is  possible  to  use  the  known  solution  of  the 
two-body  problem,  since  for  ^  =  0  equations  (9)  reduce  to  the  equations  of 


PARTICLE   ATTRACTED   BY   Tt    SPHERES. 


motion  of  a  particle  /'  when  subject  to  the  attraction  of  M  alone.  Hence, 
instead  of  the  additive  increments  /3, ,  0t ,  0, ,  /34 ,  it  is  convenient  to  introduce 
new  parameters  a,  e,  9,  <p  defined  by  the  relations 


(H) 


w-  A-arcooBf-??^!  -arc 
Ll-ecos0J 


a(i-ecos0) 

=' 
l-ecos(0-?) 


*£&=='  1, 
-?)J 


By  introducing  ^  in  the  fourth  of  equations  (14)  it  is  possible  to  use  the  two- 
body  problem  for  determining  all  terms  of  the  solution  which  are  independent 
of  n'.  In  terms  of  the  parameters  a,  e,  6,  <p  the  properties  of  the  solution 


/',  a  the  position  of  the 
particle  at  T— 0. 

IF,  is  the  longitude  of  the 
particle  at  T— 0. 

Wt  -  W.  is  the  longitude  of 
perihelion. 


W-axis 


Fiu.  8. 

are  well  known,  and  the  conditions  of  periodicity  can  be  easily  discussed. 
The  geometric  meaning  of  the  angles  6,  <p,  Wt ,  and  Wt ,  which  occur  below,  is 
shown  in  Fig.  8. 

On  making  the  substitution  w  =  u  —  nr  and  then  setting  n  =  0  in  equa- 
tions (9),  we  obtain  the  differential  equations  of  the  two-body  problem,  of 
which  the  solution  is 

r=(l+a)(l-ecos#),  '  "*     "*x       cosE-e 


sin  (u+ IP,- 


where  E  is  defined  by  the  relation 

(T+^ 


Vl—^eanE 
1—ecoeE  ' 


(15) 


394  PERIODIC   ORBITS. 

All  other  terms  in  the  solution  involve  /**.     To  find  the  terms  in  M*  and  n*<p 
we  will  write 


On  substituting  these  expressions  in  the  differential  equations,  there  results 
for  the  determination  of  p2  and  wt  the  following  set  of  equations  : 


<&  _2?  -  3P2  =         (l+3cos2T)  +          -J  (3cosr+5cos3r), 

UT  UT  ^  O       A. 

(sinr  +  5sin3r). 


These  equations  are  integrable,  and  the  result  of  integration  is 


By  a  similar  computation  it  is  shown  that 


The  terms  independent  of  //  are  obtained  from  equations  (15)  by  Taylor's 
expansion  and  the  relation  W  =  U  —  ^T;  and  the  solution  becomes 


(16) 


p=l-fa  —  ecosr  — 

r-  |TCOST) 

—  T/i  —  3aercosT  —  e0(l  —  2cosr) 


Applying  the  conditions  (10)  that  the  solution  shall  be  periodic,  we  have 

(a)  0 

(fc)  0 

(c)  0=  —  3jt>7ra  —  2pirfjL  —  6pirae-\- 

(d)  0 


(17) 


PARTICLE    ATTRACTED    BY   n    SIMI1  1  395 

The  conditions  (17)  involve  the  four  quantities  a,  e,  0,  tp,  and,  if  independent . 
would  determine  them  in  terms  of  ^.  Hut  the  differential  equations  (9) 
do  not  involve  r  explicitly  and  hence  admit  the  integral  of  Jacobi.  This 
furnishes  a  relation  of  the  type 

F(a,  e,  0,  <p,  n)  =  constant. 

and  equation-  17  are  not  independent.*  It  follows  that  if  (a),  (&),and(c) 
are  solved  for  the  three  quantities  o,  e,  and  8  in  terms  of  n  and  <p  and  the 
results  substituted  in  (d),  the  equation  is  satisfied  identically  in  <p.  In  this 
problem  the  dynamical  interpretation  is  simple.  Since  the  finite  bodies 
move  in  circles  the  origin  of  time  is  arbitrary.!  The  most  convenient 
choice  is  r  =  0  when  w  =  Q,  which  is  equivalent  to  choosing  ^>  =  0. 

Consider  the  solution  of  equations  (a),  (6),  and  (c)  for  a,  e,  and  0.  The 
equations  have  the  following  properties: 

(I)  There  are  no  terms  independent  of  a  and  /*.     This  follows  from  the 
fact  that ,  in  the  two-body  problem,  the  period  does  not  depend  upon  e  and  d. 

(II)  There  are  no  terms  involving  M  to  the  first  degree  except  the  one 
term  —  2pirn,  which  occurs  in  (c). 

(III)  There  are  no  terms  in  d  independent  of  e,  since  6  does  not  enter 
the  initial  conditions  independently  of  e.     It  follows  from  these  properties 
and  the  particular  form  of  the  first  terms  of  the  equations  that  a,  e,  and  8 
are  determined  uniquely  as  power  series  in  n  by  the  following  steps: 

(1)  From  (c)  we  obtain 

a  =  M[-f+   •  •  •  +  f unction (M,  e,  0)]. 

(2)  This  value  of  a  when  substituted  in  (6)  permits  a  factor  M  to  be 
divided  out.     We  can  then  solve  the  result  for  e  as  a  power  series  in  n  and 
6  which  contains  n  as  a  factor,  and  obtain 

e  =  4loi  +  '  '  '  +  function  („,*)]. 

(3)  When  the  values  of  a  and  e  are  substituted  in  (a)  a  factor  f  can  be 
divided  out  and  0  obtained  as  a  power  series  in  /*  alone,  vanishing  with  p. 

(4)  By  the  substitution  of  the  value  of  0  thus  found  in  the  expressions 
for  e  and  a,  we  obtain  finally 

a=Mpi(/<0,       e  =  npt(n),       0 =/#,(/*). 

The  preceding  operations  are  known  to  be  convergent  for  all  values  of 
a,  e,  6,  and  n  which  are  sufficiently  small.  Hence,  for  a  given  value  of  M 
sufficiently  small,  it  is  possible  to  determine  the  initial  conditions  (14)  as 
power  series  in  ^  such  that  the  solution  of  the  differential  equations  (9) 
shall  be  periodic  in  r  with  the  period  2pr. 

•See  Poincar6,  loe.  eit.,  p.  87. 

tWben  the  finite  bodies  do  not  form  a  fixed  configuration  in  the  rotating  plane  the  integral  of  Jacobi 
does  not  exist  and  the  origin  of  time  U  not  arbitrary.  In  thU  case  it  u  neceasary  to  determine  the  four 
parameters  from  the  conditions  of  periodicity.  The  case  of  the  triangular  solution  when  the  finite  bodiea 
move  in  ellipses  has  been  treated  by  I»ngley  in  a  paper  in  the  Tran*tctunu  of  the  American  Mathematical 
Society,  vol.  8  (1907),  PP-  159-188. 


396  PERIODIC   ORBITS. 

When  the  values  of  a,  e,  6  in  terms  of  n  are  substituted  in  equations 
(16)  the  periodic  solution  is  obtained.  The  period  of  the  solution  is  2pir 
in  T,  where  p  is  an  integer,  and  from  the  conditions  of  periodicity  (10)  it 
is  apparent  that  the  particle  makes  p  revolutions  in  the  rotating  plane 
during  a  period.  The  process  by  which  the  periodic  solution  was  obtained 
yields  a  unique  result;  therefore,  for  an  assigned  value  of  n,  there  exists  one, 
and  only  one,  orbit  having  the  period  2pir.  Since  the  orbits  having  the 
period  2pir,  p>l,  include  those  having  the  period  27r,  it  follows  that  all  the 
orbits  of  this  analytic  type  are  closed  after  one  synodic  revolution* 

Since  r  =  vt  the  period  of  the  solution  in  t  is  27r/»,  and  the  quantity  v, 
which  is  so  far  arbitrary,  can  be  determined  by  assigning  the  period  of  the 
solution.  The  parameter  ^  is  then  determined  by  the  relation  fj,  =  m=N/v; 
that  is,  the  numerical  value  of  p  is  the  ratio  of  the  mean  motions  of  the 
finite  bodies  and  of  the  particle.  If  the  direction  of  revolution  of  the  par- 
ticle is  the  same  as  that  of  the  finite  bodies — that  is,  if  the  orbit  is  direct— 
v  and  N  have  the  same  sign  and  n  is  positive;  if  the  orbit  is  retrograde,  v 
and  N  have  opposite  signs  and  /z  is  negative.  Since  for  an  assigned  value 
of  M  there  exists  one,  and  only  one,  periodic  orbit,  and  since  values  of  fj, 
which  are  numerically  equal,  but  opposite  in  sign,  give  orbits  having  the 
same  period  in  T,  it  follows  that  for  a  given  period  there  exist  two,  and  only 
two,  real  orbits  of  the  type  under  consideration.  In  one  the  motion  is 
direct,  and  in  the  other  it  is  retrograde. 

We  may  now  state  the  result  as  follows :  The  period  2ir/v  of  the  solution 
may  be  assigned  arbitrarily  in  advance,  subject  only  to  the  condition  that  the 
ratio  N/v  is  sufficiently  small,  where  2ir/N  is  the  period  of  the  motion  of  the 
finite  bodies.  Then  there  exist  two,  and  only  two,  real  periodic  orbits  of  the 
particle  having  the  required  period.  In  one  the  motion  is  direct,  and  in  the 
other  it  is  retrograde.  All  the  orbits  of  this  type  are  closed  after  one  synodic 
revolution. 

In  deriving  this  conclusion  no  use  was  made  of  the  explicit  values  of 
those  terms  in  the  disturbing  function  which  are  due  to  the  body  M2 .  The 
proof  depends  entirely  upon  the  form  of  certain  terms  of  the  solution  which 
involve  \.  Hence  the  analysis  and  conclusions  are  applicable  without 
change  to  the  case  where  n  finite  bodies  revolve  in  circles  in  such  a  way  as 
to  form  in  the  rotating  plane  a  fixed  configuration. 

199.  Construction  of  Periodic  Orbits  Having  no  Line  of  Symmetry. — It 
is  possible  to  construct  the  periodic  solutions  of  the  differential  equations 
(9)  by  the  method  indicated  in  the  existence  proof,  but  the  process  is  labo- 
rious. A  method  will  now  be  given  by  which  the  solution  to  any  desired 
number  of  terms  can  be  conveniently  constructed.  It  is  not  necessary  to 
determine  the  initial  conditions  explicitly  in  advance,  and  the  computation 
involves  only  algebraic  processes. 

*Since  no  new  orbits  are  obtained  by  taking  p>  1  we  will  assume  hereafter  that  p  =  1. 


PARTICLE   ATTRACTED    BY   n   SPHERES.  397 

It  has  been  proved  that  the  periodic  solutions  are  expressible  in  the 
form 

-     =  1  *         '    '    '  *         '  ;  •  i 

I      ('8) 


The  series  (18)  satisfy  the  differential  equations  (9)  uniformly  over  a  finite 
interval  in  n,  and  hence,  when  the  series  are  substituted  in  the  differential 
equations,  the  coefficient  of  each  power  of  p  must  vanish.  Furthermore, 
the  series  are  periodic  with  period  2*  in  r;  and,  because  the  periodicity 
holds  for  a  continuous  range  of  values  of  n,  each  coefficient  p,  and  wt  sepa- 
rately is  {xriodic  with  the  period  2r  in  r.  It  has  been  shown  also  that  we 
can  choose  w  =  0  when  r  =  Q,  and  because  this  holds  identically  in  ft,  it  follows 
that  wt  (0)  =  0  for  every  t. 

Let  the  solution  (18)  be  substituted  in  the  differential  equations  (9) 
and  arrange  the  results  as  power  series  in  ft.  The  terms  of  the  first  members 
have  the  following  forms,  where  the  accents  indicate  derivatives  with 
respect  to  r: 


+  2pX-.+ 


-+••'•  +K+PX-.+  •  •  •  +P.-XV+ 


(19) 


398  PERIODIC   ORBITS. 

The  second  members  have  no  terms  independent  of  M2-  Therefore,  on 
equating  to  zero  the  coefficients  of  the  first  power  of  n,  we  have  for  the 
determination  of  pl  and  wl 

pi_n  xom 

~ 


It  follows  from  these  equations  that 

Pl  =  2  (  1  +  O  +  c<"  cos  T  +  c™  sin  r,  1 

where  cj",  c^",  c(",  c("  are  constants  of  integration.  Since  Pl  and  w^  are 
periodic,  the  coefficient  of  T  in  w^  must  vanish.  This  condition  determines 
the  constant  cf,  namely,  c™=  —4/3.  Since  w\  =  0  when  r  =  0,  C40=  —  2c"'. 
The  constants  c,"  and  c"'  are  so  far  undetermined. 

On  equating  to  zero  the  coefficients  of  the  second  power  of  /*,  the  fol- 
lowing set  of  equations  is  obtained  : 


<fwz 

"~Ve» 


where  /0  and  </„  are  obtained  from  /  and  g  respectively  by  writing  /x  =  0, 
W  =  T,  p=l.  The  second  members  are  known  functions  of  T  and  the  equa- 
tions are  explicitly 


cos3r,          ,    (23) 
=  (-| c"  +  DJ2' )  sin  T  —  ^Cj"  cos  T + D®  sin  2r  +Z>j2)  sin  3T. 


On  integrating  the  second  equation,  we  have 

-cos3r.     (24) 


On  eliminating  -p2  from  the  first  of  equations  (23)  by  means  of  equation 
(24),  there  results 

s  -*^ 

/\  I    O  xi^*'     I    I 2  /^         I     yi  '  '  ^_  O  If        I  f*O^  T*     -    x  /*       ^1T1  T 

(25) 


-  |Z)®)cos3r. 


PARTICLE   ATTRACTED   BY   Tl   SPHERES.  399 

In  order  that  the  solution  of  equations  (25)  shall  contain  no  non-periodic 
term,  the  coefficients  of  COST  and  sinr  must  vanish;  hence  ri"  and  c{"  are 
determined  by  the  conditions 


With  these  values  of  c™  and  c"'  the  solution  becomes 

P,  =  4 
where 


On  substituting  this  value  of  p,  in  equation  (24)  and  integrating,  we  obtain 
for  u>,  a  solution  of  the  form 

Wt  =  cf  -  (2.C+3O  r-2cJI>8inT+2c>cl)cosT+«;i)  8in2r+^a>sin3r, 
where 

6?  =  -  4  (D?+2jti?)  0'-2,  3). 

Since  wt  is  periodic,  c™  is  determined  by  the  condition 

2  A?  +3<f-0. 

Since  wt  =  0  when  r  =  0,  c™  is  expressible  in  terms  of  c}1',  namely, 

cf  +2cf-0. 

Of  the  eight  constants  of  integration  which  have  been  introduced  in  the 
first  two  steps,  five  (c,a>,  cjn,  cj",  ci",  c™)  have  been  determined  uniquely; 
ej"  has  been  expressed  uniquely  in  terms  of  cf  ;  while  the  remaining  two 
(c?,  O  are  still  arbitrary. 

By  equating  to  zero  the  coefficients  of  the  third  power  of  M  the  following 
set  of  equations  is  obtained: 


(96) 


where  /,  and  </,  denote  the  coefficients  of  n  in  /  and  g  respectively.    The 
second  members  are  known  functions  of  r,  and  the  equations  have  the  form 


+Af  COBJT+B? Binjr  0'-2,  3,  4). 

+D?mnJT+C?coBJT. 


400  PERIODIC   ORBITS. 

The  treatment  of  equations  (27)  proceeds  by  steps  similar  to  those 
employed  in  the  solution  of  equations  (23).  Four  new  constants  of  inte- 
gration are  introduced,  namely,  cf\  cf,  cf,  cf,  while  cf,  cf,  and  cf  are 
uniquely  determined  by  the  conditions 


The  solution  has  the  form 


From  the  condition  that  w3  —  0  when  T  =  0,  it  follows  that  cf  is  expres- 
sible uniquely  in  terms  of  cf  by  the  relation 

<f+2c«)+  S  7f  =  0. 

J=2 

The  two  constants  c£"  and  c,3)  are  determined  in  the  next  step. 

It  can  be  established  by  complete  induction  that  the  preceding  process 
can  be  carried  as  far  as  is  desired.  Suppose  p,,  wv]  pu  wt;  .  .  .  ;  pi^l,  w(^ 
have  been  determined  by  this  process.  The  expressions  have  the  following 
form: 

ft  =  C  +  JS  (a?  cosjr  +  j8«  sinjV)  » 

1+1    /  v 

Wt  =  Ti°  +  2i  («f  dtojr+yf  COS  JT)  (i=  1,  2,  .  .  .  ,  i-2), 

P4_,  =  a"'"  +G;'-"  cos  r  +C?-1'  sin  r  +  S  (o«-°  cos  JT  +  ff~°  sinjr)  > 

^^B^ 


+  S  (ay-"  siiyV  +  7?-° 


The  constants  of  integration  have  been  uniquely  determined  except  c""", 
c?~",  and  ci'~".  The  first  two  are  so  far  arbitrary,  while  c"""  is  expressible 
in  terms  of  c£~u  by  the  relation 


PAUTICLK    ATTRACTED    BY    n   SI'lll  401 

The  equations  for  the  determination  of  p,  and  w,  have  the  form 


1+1 

+  S 

CJS) 


Thr  cuetlicients  A,  B,  C,  D  arc  known  constants  and  equations  (28)  are 
solved  \}\  the  steps  employed  in  the  solution  of  equations  (23).  During  the 
process  four  constants  of  integration  are  introduced,  namely  c{'\  cj°,  cj°,  c"', 
and  four  are  uniquely  determined  by  the  conditions 


?-"=  -2ci'-°-  S  T)''",  3<f=  - 


i-\ 


(29) 


The  solution  of  equations  (28)  is 

1+1  /  \ 

inr+  S  ( aJ°cos./T  +  /3/0sin</Tj, 

(30) 


where  the  coefficients  are  given  by  the  formulas 


„ 
O, 


«>_        1 
=  —     - 


(0  —  -     (n---(n 


(31) 


The  formulas  (31)  together  with  the  conditions  (29)  are  sufficient  to 
construct  the  periodic  solution  of  equations  (9)  to  any  desired  degree  of 
accuracy;  the  computation  is  entirely  algebraic.  In  order  to  determine  the 
constants  of  integration  entering  in  the  last  (Ith)  step,  it  is  necessary  to 
compute  the  coefficients  ^|l+l),  fl^",  CT",  D?+"  of  the  next  following  step. 


402  PERIODIC   ORBITS. 

200.  Numerical  Example  1.— For  the  purpose  of  illustration,  we  assign 
numbers  to  the  constants  involved  in  the  preceding  analysis  and  construct 
an  orbit.  In  this  and  the  other  numerical  examples  which  occur  later,  it  has 
not  been  shown  that  the  processes  are  valid  for  the  numerical  values  which 
are  employed  and  which  have  been  selected  for  convenience  in  graphical 
representation.  It  is  probable  that  the  series  are  convergent,  although  it 
has  not  been  found  possible  to  determine  the  true  radii  of  convergence. 

The  differential  equations  of  motion  are  equations  (5).  On  putting 
ra  =  M  and  writing  the  second  members  explicitly  as  far  as  terms  of  the 
second  degree  in  a/A,  we  have 


;~i  )+35cos4(>-J)]+ 


(32) 
d*w 


We  select  M  for  the  unit  of  mass  and  suppose  Ml  =  10,  M2  =  5.  For  the  unit 
of  distance  we  take  the  distance  between  the  finite  bodies,  that  is,  A  =  l; 
and  the  unit  of  time  is  selected  so  that  N=l.  The  period  of  the  solution  is 
assigned  so  that  v  =  5,  whence 


PAUTICLK    ATTUACTKI)    BY    tl    8PHKHKS. 

The  constant  A"  is  detenniaed  from  the  relation 

#'/!'  =  A'l.U  +  A/.+A/,), 
whence 

-0.06250,        A-1  A/,  =  0.62500,        ^,=  1.56250^. 

The  constant  a  was  defined  by  i»*a'  =  fc*Af,  whence 


\Yith  these  numerical  values  equations  (32)  become 


f  -t  =  (0.31250+0.93750cos2«?)pM' 
P 

+  (0.78125-1. 17188  cos2u>+2.02977sin2tt>)pM' 

+  (0.47714cosu>+0.79523cos3u>)pV 

+  (0.59643  cos  w>+ 1 .03308  sin  w  - 1 .98808  cos  3u>)pV 

+  (0.16188+0.35974cos2u>+0.62954cos4u>)pV+- 
(4*£  +„)  =  -  (0.93750sin2w)pM1 


+  ( 1 . 1 7 1 88  sin  2w + 2.02977  cos  2u>)  pM' 

-  (0.159058inu?+0.79523sin3u>)pV 

+  (-0.19881sinu>+0.34436cosu)+1.98808sin3tc)pV4 

-  (0.1 7987  sin  2^+0.62954  sin  4  u;)pV+ 

The  periodic  solution  of  equations  (33)  is 
p=  1  --/x+ (0.45139+0.39762  COST -0.62500  cos2T)MJ 

3 

+  ( -0.47647+ 1.04542  cosT+0.86090sinr+ 0.46875  cos  2r 
-1.35318sin2T-0.16567cos3TV+  •'••, 

-0.79524sinT+0.85938sin2T)M* 

+  (0.13882-3.25720sinT+1.72180cosT+0.27995sin2T 

-1.86062cos2T+0.19881sin3T)M'+  •••< 


(33) 


404 


PERIODIC   ORBITS. 


Substituting  the  numerical  value  /x  =  0.2,  equations  (34),  if  convergent, 
are  the  equations  of  motion  of  the  particle  P.  The  orbit  is  shown  in  Fig.  9. 
In  this  and  the  figures  of  the  following  numerical  examples  the  comparison 
circles  are  not  the  circular  orbits  which  have  been  called  the  undisturbed 
orbits.  The  undisturbed  orbits  are  referred  to  fixed  axes  while  the  draw- 
ings are  made  with  reference  to  rotating  axes.  The  comparison  circles 
represent  orbits  in  which  the  particle  would  make  a  complete  revolution 


with  respect  to  the  rotating  axes  during  the  period.  The  points  which  are 
numbered  1,  2,  .  .  .  ,8  represent  positions  of  the  particle  in  the  periodic 
orbit  at  intervals  of  r  =  w/4.  The  corresponding  positions  in  the  comparison 
circle  are  indicated  by  the  numbers  1',  2',  .  .  .  ,  8'. 


201.  Some  Particular  Solutions  of  the  Problem  of  n  Bodies. — The 
existence  of  symmetrical  periodic  orbits  of  the  particle  depends  upon  the 
masses  and  motion  of  the  finite  bodies.  So  far  as  the  analysis  is  concerned, 
this  motion  may  be  arbitrarily  periodic,  without  reference  to  the  nature 
of  the  forces  producing  it.  It  is  required  only  that  the  motion  of  the  finite 
bodies  shall  be  known  and  that  they  shall  attract  the  particle  according  to 
the  Newtonian  law.  It  will  be  interesting,  however,  in  developing  the 
analysis,  to  prescribe  motion  for  the  finite  bodies,  which  is  possible  under 
the  law  of  the  inverse  square  of  the  distance.  For  three  finite  bodies  the 
two  solutions  of  Lagrange  are  well  known.  In  the  case  of  the  equilateral- 
triangle  solution  the  periodic  orbits  of  the  particle  about  one  of  the  bodies 


PARTICLE    ATTRACTED    BY   n   SPHERES.  405 

have  a  line  of  symmetry  if  the  other  two  masses  are  equal.  In  the  case  of 
the  straight-line  solution  the  periodic  orbits  of  the  particle  about  any  one  of 
the  bodies  are  symmetrical  with  respect  to  the  line  joining  the  finite  bodies. 
The  particular  solutions  which  Lagrange  has  given  for  three  bodies  have 
been  extended  to  some  cases  of  more  than  three  bodies,*  and  we  shall 
consider  two  examples:  (1)  in  which  there  are  five  bodies,  and  (2)  in  which 
there  are  nine  bodies. 

Let  the  masses  of  n  finite  bodies  be  represented  by  Mlt  Mt,  .  .  .  ,Mm. 
Suppose  that  the  bodies  lie  always  in  the  same  plane,  and  that  their  coordi- 
nates with  respect  to  their  common  center  of  mass  as  origin  and  a  system  of 
rectangular  axes  which  rotate  with  the  uniform  angular  velocity  N  are, 
respectively,  (xlt  y,),  (xt,  j/t),  .  .  .  ,  (x,,yn).  Supposing  that  the  bodies 
attract  each  other  according  to  the  Newtonian  law,  the  differential  equations 
of  motion  are 


(35) 


+  2  N      -  JVfc  =  - 
ar  at 


V(xl-Xj)t+(yt-y)y  (»-l,  2,  ....  n;  J 


If  we  assume  that  each  body  is  revolving  in  a  circle  about  the  common 
center  of  mass  of  the  system  with  the  uniform  angular  velocity  N,  its  coordi- 
nates with  respect  to  the  rotating  axes  are  constants  and  the  derivatives  of 
the  coordinates  with  respect  to  the  time  are  zero.  Equations  (35)  therefore 
reduce  to  the  following  system  of  algebraic  equations 


=0 


(36) 


It  follows  from  these  equations  that 

Mlxl+Msxt+  •  •  •  -fA/,,x.  =  0,      Af.T/.-f-Af.t/.-r-  •  •  •  +^.  =  0,    (37) 

which  express  the  fact  that  the  origin  of  coordinates  is  at  the  center  of  mass. 
These  equations  may  be  used  instead  of  two  of  (36). 

•See  Hoppe,  "Krweiterung  der  bekannlen  Special  loaungen  dec  Dnskorperprubleins;"  Arekit.der  Matk. 
tmd  P*»».,  vol.  64,  pp.  218-223.     Andoyer.  "Sur  l'<quilibn  rolatif  de  n  corp.;"  Bu«.  «<ron.,  vol.  23 
pp.  50-^59.     Longley,  "Sonic  particular  solutions  in  the  problem  of  n  Ixxlies;     Bull.  Amtr.  M  M.  Aoc.,  vol. 
13  (1907),  pp.  324-336. 


406  PERIODIC   ORBITS. 

This  system  of  2n  simultaneous  algebraic  equations  involves  the  square 
of  the  angular  velocity,  N,  the  n—1  ratios  of  the  masses,  and  the  2w-l 
ratios  of  the  distances  xt  and  yt  .  Accordingly  w  —  1  of  these  quantities  may 
be  chosen  arbitrarily  and,  if  the  resulting  equations  are  independent,  the 
remaining  In  quantities  are  determined  by  the  relations  (36).  In  order  to 
admit  physical  interpretation,  the  quantity  N2  and  the  masses  must  be  real 
and  positive,  while  the  coordinates  must  be  real.  With  these  restrictions 
it  is  not  easy  to  discuss  the  general  solutions  of  equations  (36),  but  some 
interesting  results  can  be  obtained  by  a  study  of  special  cases.  If,  in  the 
problem  of  three  bodies,  the  assumption  is  made  that  the  triangle  formed 
by  the  three  bodies  is  isosceles,  it  can  be  shown  that  equations  (36)  can  be 
satisfied  only  if  the  triangle  is  also  equilateral. 

On  supposing  the  number  of  bodies  to  be  five,  the  system  of  equations 
to  be  satisfied  is 


(a) 


l  ~  X*  -4- 

~3  ~l 


~3  ~l  ~3 

'1.4  '1,5 


'1.1  'l,3  '1.4  '1,5 


,      tx,-x     ,,x2-xt    .      6-x5  _n 

~s  —  p—  —15  ---  U, 


~s 
'2,1  's,s  '2,4 


.txt-x     ,tx,-x     , 

Ij—  3~ 


L.2-3  -3 

'3,1  'j.J  '3,4  '3.5 

W    -4-  Mtfo-s,)  ,  M2(x4-x,)  .  M3(x4-x3)  . 

ti*«»  "          3  ~*  3 


'4,J  '4,3  ^U 

(/)  <38) 


-u+    *yi  -  ft  4.    *yi  -  y»  _L        -  y    < 
\yj  i/i  i        I*  -  —  zj  — 


*  -^  —  zj  —  —~a  — 

'!.*  '1.3  '1.4  '1,5 

y«~yi)  +  M3(y,-y3}  ,  M4(y,-y4)  ,  Af.fa- 

-  ~ 


fV)       _  ff2,,  4.  M^y,  -  y,)  ,  M2(i/3  -  yt)      M  4(y,  -  y4)  ,  M6(j/3  -  y.)  _  n 

V*^  JL2  J/3    I     "  r3  ^  ~^3  ^3  -  —  U 

'3,1  '3,J  '3,4  <3,i 

fj)       -W     .M^-yQ  ,M1(y4-y1)  ,  M3(y<-y3)  ,Mt(yt-yJ_n 

\JS  p  i/4    I  ,J  Xj  ~^3  ~^3  -  —  U 

74,1  '4,Z  '4,3  'J.S 

Let  us  suppose  that  M6  lies  at  the  origin  of  coordinates  and  that  the 
other  four  masses,  which  are  equal  in  pairs,  lie  on  the  coordinate  axes  at 


PARTICLE    ATTRACTED    BY    tt   SPHERES.  407 

the  vertices  of  a  rhombus  wit  h  the  equal  masses  opposite  cadi  other  (see  Fig. 
10).     This  is  equivalent  to  the  relations 

X4"°'  Vl=  M    (39) 

-.       _         f\  -  .       __^      Tf    A  Q.        f\  JT»     J  f\  mj  *  f  •m   gg        I  \  / 

On  substituting  the  assumed  values  (39)  in  equations  (38),  we  find  that 


Fio.  10. 

(a)>  (c),  (e),  (/),  (0),  and  (t)  are  satisfied.     Equations  (6)  and  (d)  become 
identical,  yielding 

1?  =  (2^7  +  A'(  Vl+K*)'  +  ^ ' 
and  equations  (h)  and  (j)  become  identical,  yielding 
JV  2Af'  2Af"  M, 


(41) 


When  M ',  M",  Mt ,  K,  and  A  have  been  chosen  or  determined,  these  equa- 
tions insure  a  positive  value  for  N*. 

On  eliminating  N*/k?  between  equations  (40)  and  (41),  we  obtain  for  the 
relation  between  the  masses, 


M' 


(42) 


The  choice  of  the  constant  K,  which  is  the  ratio  of  the  diagonals  of  the 
rhombus,  is  limited  by  the  condition  that  the  resulting  ratio  of  the  masses 
must  be  positive.  To  investigate  this  condition  we  set  A/t=l.  Then, 
regarding  K  as  a  parameter,  equation  (42)  represents  a  straight  line  in  the 


408 


PERIODIC   ORBITS. 


M'M"  plane.  Only  those  pairs  of  values  (M',M")  which  represent  a 
point  in  the  first  quadrant  are  admissible.  This  condition  will  certainly  be 
satisfied  if  the  slope  is  positive,  that  is,  if  the  coefficient  of  M'  is  positive. 
This  condition  is  easily  found  to  be 


<K<V3. 


(43) 


If  the  slope  of  the  line  is  negative,  it  may  still  lie  partly  in  the  first  quadrant 
if  the  intercept  on  the  AT-axis  is  positive — that  is,  if  the  coefficient  of  M6 
in  equation  (42)  is  positive.  It  is  easily  verified,  however,  that  values  of 


FlQ.    11. 


K  which  make  the  slope  negative,  also  make  the  intercept  on  the  M"-axis 
negative.  Hence  the  choice  of  the  ratio  of  the  diagonals  is  limited  by  the 
condition  (43) ;  otherwise  it  is  arbitrary. 

The  conditions  for  the  rhombus  configuration  with  a  fifth  body  at  the 
center  may  be  summarized  as  follows: 

Suppose  a  value  of  K  satisfying  condition  (43)  is  assigned;  then  two  of  the 
masses  M',  M",  and  Mh  can  be  chosen  arbitrarily*  and  the  third  is  determined  by 
equation  (42).  The  length,  A,  of  one  semi-diagonal  can  be  selected  at  pleasure 
and  the  angular  velocity  is  then  determined  by  equation  (40)  or  equation  (41). 

In  the  following  discussions  this  configuration  will  be  referred  to  as 
configuration  (A). 

The  second  configuration,  (B),  will  consist  of  nine  bodies,  arranged  as 
shown  in  Fig.  11.  Four  bodies  M1}  M2,  M3,  Mt  lie  on  the  coordinate 
axes  and  on  the  circumference  of  a  circle  of  radius  A;  four  others,  M6,  M, 


t 1 


*Except  when  K  =  1  and  the  rhombus  becomes  a  square.     Then  M '  and  M"  must  be  equal. 


PAKTICLE   ATTRACTED    BY   Tl   SPHERES.  409 

.V. ,  Af,  lie  on  tin-  bisectors  of  the  angles  between  the  coordinate  axes  and 
on  a  circle  of  radius  AM,  while  the  ninth  body  is  at  the  center.  Supposing 
that  all  the  masses  (»n  (he  same  circle  arc  equal,  we  have  the  following  con- 
ditions: 


(44) 


( 'orres|x>nding  to  equations  (38)  of  the  five-body  problem,  there  is  a  set 
of  equations,  which,  upon  the  assumptions  (44),  reduce  to 

K 


45) 


Eliminating  N*Al/l£  from  equations  (45),  there  results  the  following 
condition  on  the  masses  A/',  A/',  A/,,  and  the  ratio,  K,  of  the  radii  of  the 
circles, 


M,       |      1 2Af 

V  \W^4j^ 


(46) 

-        -  - 

M/l^i~4~ 


The  choice  of  the  ratio  K  and  two  of  the  masses  is  limited  by  the  condi- 
tions that  the  third  mass,  as  determined  by  equation  (46)  and  the  square 
of  the  angular  velocity,  N',  from  (45),  shall  be  positive.  The  limits  within 
which  the  choice  can  be  made  have  not  been  determined,  but  for  the  purpose 
of  application  in  numerical  example  3,  the  following  set  of  values  satisfying 
the  conditions  has  been  computed: 

A'  =  2,      A/'  =  l,      A  =  l,     Af.  =  l,     A/'  =  8.2526,     #'  =  1.6399**. 


410  PERIODIC   ORBITS. 

202.  Existence  of  Symmetrical  Periodic  Orbits.  —  For  the  development 
of  the  type  of  analysis  applicable  to  symmetrical  orbits  we  shall  use  configu- 
ration (^4.)  of  the  preceding  section,  the  notation  being  unchanged  except 
that  the  mass  at  the  center  will  now  be  denoted  by  M  instead  of  M5  .  With 
reference  to  M  as  origin  and  an  axis  having  a  fixed  direction  in  space,  let 
the  polar  coordinates  of  Mlt  M2,  M3,  M4,  and  P  be,  respectively,  (/2UF,), 
(#2,F2),  (#3,F3),  (#4,F4),  and  (r,  v).  The  coordinates  of  the  bodies  are 
expressed  in  terms  of  the  time  as  follows: 


where  N  is  given  by  equation  (40)  or  equation  (41). 
The  differential  equations  of  motion  of  P  are 

r  dv 


(A~ 
where 


T\  T%  7*3  7*4 


v-  F2)  -  §rcos(z>-  F3)  -  jj±rcos(v-  F4) 


-2rA  cos(v-  FO,         r3=  Vi*+A'-2rAcos(v-  F3), 


r*  =  Vrt+A2-2rKAcos(v-  F,),       r,=  Vr*+K*A*-2rKAcos(v-  F4). 
We  now  define  m  and  a  by  the  relations 


where,  as  in  the  preceding  case,  v  denotes  the  mean  angular  velocity  of  P. 
The  motion  is  referred  to  an  axis  rotating  with  the  angular  velocity  N  and 
passing  always  through  Ml  by  the  substitution  v  =  w-\-Vl  =  w+Nt,  and 
factors  depending  on  the  units  employed  are  eliminated  by  the  substitution 
r  =  ap,  vt  =  T.  We  obtain  then  the  differential  equations  of  relative  motion 

^V^.    (48) 


These  equations  have  the  same  form  as  the  set  (9)  and  the  analysis 
and  results  of  that  problem  are  applicable  in  this  case.  We  know,  then, 
that  there  exists  one,  and  only  one,  orbit  in  which  the  particle  P  moves 
with  direct  motion  and  with  a  preassigned  period.  We  shall  see  that  there 
exists  one,  and  only  one,  such  orbit  which  is  symmetrical  to  the  line  joining 
M  and  Ml  (also  to  the  line  joining  M  and  Ms)  ;  hence  it  will  follow  that  there 
are  no  unsymmetrical  orbits  of  this  type.  Furthermore,  all  the  periodic 
orbits  which  are  given  by  this  analysis  are  closed  after  one  revolution  in 
the  rotating  plane;  hence  in  case  symmetrical  orbits  exist,  they  are  also  closed 
after  one  revolution. 


PARTICLE    ATTRACTED    BY    T»    SPHERES.  411 

We  can  expand  S2  as  a  power  series  in  up/A  which  i<  convergent  fur  all 
values  of  w  so  long  as  the  distance,  ap,  of  tin-  panicle  from  M  remains  less 
than  the  distance  from  M  to  the  nearest  finite  body;  and  in  all  that  follow.- 
this  condition  is  supposed  to  he  satisfied.  The  expansion  has  the  form 


i(jp)'|3cos(to-r)+5cos3(tr-r)}+ 


From  the  conditions  of  the  configuration  (A)  we  have 


Let  X,  and  X,  be  defined  by  the  relations 

\ 


=  X'V1"  ""(V 


From  equations  (40)  we  have 


On  setting  the  coefficient  of  N*  in  this  equation  equal  to  «,  it  follows  that 
the  second  members  of  equations  (48)  have  the  form 

-I-  J*L  =  <cw'p[XI(l+3cos2u>)+Xi(l-3cos2uO 

+ terms  involving  only  cosines  of  even  multiples  of  tc], 

(49) 

-±-  ^  =  -Kmtpr3Xlsin2u)-3Xtsin2u; 
ftato  dw 


+  terms  involving  only  sines  of  even  multiples  of  w\. 


412  PERIODIC   ORBITS. 

On  introducing  the  parameter  of  j  integration  n  by  the  relations  m  =  n 
and  a/ A  =  TJ/Z,  where  T\  is  a  numerical  constant,  equations  (48)  become 


/dw 

d?-p(d^ 

where 

/=Kp[X1(l+3cos2w)+X2(l-3cos2w/)+  •  •  •  ], 
g=  —  Kp[3(\  —  X2)sin2w+  •  •  •  ]• 
Suppose 

P  =  *I(T),         W  =  ^(T)  (51) 

is  a  solution  of  equations  (50)  such  that  p'  =  w  =  0  at  r  =  0.  Then  it  follows 
from  the  form  of  the  differential  equations  that  ^,  is  an  even  function,  and 
that  ^8  is  an  odd  function  of  T.  Hence,  if  the  particle  crosses  the  w-axis 
orthogonally,  the  orbit  is  symmetrical  with  respect  to  the  line  w  =  0,  and 
with  respect  to  the  time  of  crossing  this  line.*  Suppose  that  when  T  =  TT  the 
particle  crosses  this  line  (or,  what  is  the  same  thing,  the  line  w  =  T)  again 
orthogonally;  the  orbit  will  be  symmetrical  with  respect  to  this  line  and  the 
time  of  crossing,  and  the  particle  will  have  again  its  initial  position  and 
relative  components  of  velocity  at  the  end  of  the  period  T  =  2-ir.  Hence 
sufficient  conditions  that  the  solution  (51)  shall  be  periodic  are 

p'(7r)=0,         «;(*•)- IT  =  0.  (52) 

For  fj.  =  0  the  equations  have  the  form  which  occurs  in  the  problem  of 
two  bodies,  and  a  symmetrical  solution  having  the  required  period  is  known, 
namely, 

p=l,        W  =  T. 

The  initial  conditions  for  this  solution  are 

p  =  l,        p'  =  0,        w  =  0,        w'  =  \. 

Consider  the  solution  for  values  of  n  different  from  zero  but  sufficiently 
small,  and  let  the  initial  conditions  for  T  =  0  be 


(53) 


*It  may  be  remarked  also,  in  this  particular  example,  that  if  the  solution  (51)  is  subject  to  the  initial 
conditions,  p'=0,  w=jr/2,  (T  =  TI),  then  1^1  is  an  even  function,  and  4/,  —  jr/2  is  an  odd  function  of  T  —  T, 
and  hence,  if  the  particle  crosses  the  line  w  —  r/2  orthogonally,  the  orbit  is  symmetrical  to  this  line  and 
the  epoch  T  =  TI.  The  sufficient  conditions  of  periodicity  (52)  might  be  replaced  by  the  conditions 


since  the  orbit  would  then  have  two  lines  of  symmetry.    For  the  purpose  of  covering  more  general  cases  it 
is  better  to  base  the  existence  proof  on  only  one  line  of  symmetry. 


PARTICLE    ATTRACTED   BY    n   SPHERES.  413 

The  solution  i.s  symmetrical  with  respect  to  tho  epoch  T  =  0,  and  can  be 
expressed  as  power  series  in  o,  r,  and  M.  which  arc  convergent  for  an  interval 
in  T  including  the  interval  0  to  *-,  if  the  parameters  are  sufficiently  small. 
If  a  and  6  ean  l>e  determined  in  terms  of  /u,  vanishing  with  /i,  so  that  the 
conditions  (52)  are  satisfied,  then  the  solution  will  be  periodic  with  the 
period  2w.  All  terms  of  the  solution  which  are  independent  of  M*  can  be 
obtained  from  the  two-body  problem  by  making  the  substitution  w  =  u  —  pr. 
These  terms  are  given  in  finite  form  by  the  expressions 


(  C08g~cj)  = 
\l-ecosES 


=  arccos  =  arc  sn 


l— 

where  E  is  defined  by  the  relation 


(  )n  returning  to  the  variable  w,  writing  the  terms  in  a  and  e  as  power 
series  by  Taylor's  expansion,  and  applying  the  conditions  (52),  we  obtain 
the  equations 

0=-fTae+   •   •  •  ,  0=-fTa-TM+   •   •  •  •  (54) 

It  follows  from  the  known  properties  of  the  series  that  there  are  no  terms  in 
e  alone,  and  there  are  no  terms  involving  ju  to  the  first  degree  except  the 
term  —  TM-  The  equations  are  satisfied  by  a  =  e  =  n  =  0,  and  in  the  second 
the  coefficient  of  the  first  power  of  a  is  not  zero;  hence  the  second  equation 
can  be  solved  uniquely  for  a  as  a  power  series  in  e  and  n,  which  contains  n  as 
a  factor.  The  result  has  the  form 

(55) 

When  this  value  of  a  is  substituted  in  the  first  of  equations  (54),  a  factor  M 
can  be  divided  out,  leaving  0  =  ire+  •  •  •  •  This  equation  is  satisfied 
by  e  =  M  =  0,  and  since  the  coefficient  of  the  first  power  of  e  is  not  zero,  it 
furnishes  a  unique  determination  of  e  in  terms  of  n,  vanishing  with  n.  When 
this  value  of  e  is  substituted  in  equation  (55),  we  have  o  expressed  uniquely 
in  terms  of  /*,  vanishing  with  n. 

Hence  for  a  given  value  of  n  sufficiently  small  it  is  possible  to  determine 
the  initial  conditions  (53)  as  power  series  in  n  such  that  the  solution  in  n  is 
symmetrical  with  respect  to  the  line  w=*  and  the  epoch  T  —  -K.  Since  it  is  sym- 
tn<  Irical  also  with  respect  to  the  line  w  =  0  and  the  epoch  r  =  0,  it  is  periodic  in  T 
with  period  IT. 

The  orbit  is  symmetrical  with  respect  to  the  line  joining  the  bodies 
M  and  M,.  If  we  take  for  the  initial  line  the  line  joining  M  and  Af,,  it 
follows  from  the  same  analysis  that  the  orbit  is  symmetrical  with  respect  to 


414  PERIODIC   ORBITS. 

this  line  and  the  time  of  crossing  it.  Since  for  a  given  value  of  /x  there  is 
only  one  periodic  orbit  of  this  type,  it  follows  that  it  is  symmetrical  with 
respect  to  both  the  lines  joining  M  with  Ml  and  with  Mt  .  For  the  con- 
figuration (A)  of  the  finite  bodies  the  periodic  orbits  of  the  particle  have  two 
lines  of  symmetry;  there  are  four  apses  and  the  apsidal  angle  is  ir/2.  For 
the  configuration  (B)  (see  numerical  example  3)  the  periodic  orbits  have 
four  lines  of  symmetry,  there  are  eight  apses,  and  the  apsidal  angle  is  jr/4 

In  establishing  the  uniqueness  of  the  periodic  solution  of  equations  (54) 
it  is  to  be  noted  that  no  use  was  made  of  the  explicit  values  of  the  terms 
from  the  second  members  of  equations  (50).  Hence,  if  the  second  members 
have  forms  which  permit  symmetrical  solutions,  the  preceding  analysis  is 
applicable  without  change  to  show  the  existence  of  symmetrical  periodic 
orbits.  //  the  masses  and  motions  of  the  finite  bodies  are  such  that  there  can 
exist  orbits  of  the  particle  about  one  of  the  bodies  having  a  line  of  symmetry,  we 
have  established  the  existence  of  periodic  orbits  having  this  line  of  symmetry. 

As  in  the  problem  treated  in  §  198,  the  period  of  the  solution  in  t  may  be 
assigned  arbitrarily  (that  is,  when  the  finite  bodies  form  a  fixed  configuration 
in  the  rotating  plane).  There  exist  then  two,  and  only  two,  symmetrical 
closed  orbits  having  the  required  period;  in  one  the  motion  is  direct,  and 
in  the  other  it  is  retrograde. 

203.  Construction  of  Symmetrical  Periodic  Orbits.—  The  method  of 
constructing  symmetrical  periodic  solutions  is  similar  to  that  explained  in 
§199.  There  is  a  slight  difference  in  the  conditions  which  determine  the 
constants  of  integration,  and  the  calculation  is  simpler  because  p  contains 
only  cosines  of  multiples  of  r,  while  w  contains  only  sines.  It  has  been 
proved  that  symmetrical  periodic  solutions  of  equations  (50)  exist,  and  that 
they  are  expressible  in  the  form  (18). 

Since  the  solution  is  periodic  for  a  continuous  range  of  values  of  /i,  each 
coefficient  p,  and  wt  is  periodic  with  period  2  w  in  T.  Also,  since  the  initial 
conditions  p'(0)  =  0,  w(0)  =  0  hold  identically  in  p.,  it  follows  that  p^(0)  =0, 
and  wt(0)  =  0  for  every  i. 

The  left  members  of  equations  (50)  are  the  same  as  the  left  members 
of  equations  (9),  and  therefore,  when  the  solution  (18)  is  substituted  in 
equations  (50),  the  terms  of  the  left  members  have  the  form  (19).  The 
right  members  have  no  terms  independent  of  ju2>  and  the  equations  for  the 
determination  of  the  coefficients  of  the  first  power  of  n  are 


of  which  the  solution  is 

+cjl))+^1)cosr+^)sinr, 

i))T-2c;i)  sinr+2<f  COST. 


PARTICLE    ATTRACTED    BY    n    SPHERES.  415 

Since  »/•,  is  |*>riodic  the  coefficient  of  T  must  be  zero;  whence  rl"  =  -4/3. 
The  constants  c"'  and  f{"  arc  determined  l»y  the  conditions 

Pi(0)=0,         ir,(0)-0. 

Therefore  rj"  =  c?  =  0.     The  constant  ri"  is  determined  in  the  following 
step  of  the  integration  [see  equations  (23)]. 

This  process  is  applicable  to  all  the  succeeding  steps.  The  differential 
equations  (50)  have  a  particular  form  which  admits  a  symmetrical  solution, 
and  it  can  be  established  by  complete  induction  that  the  equations  for  the 
determination  of  p,  and  «',  have  the  form 


(fr1 

(56) 

P|     indpt   _ 


where  c"~"  is  determined  by  the  condition  [compare  equations  (29)] 


The  solution  of  equations  (56)  is 

"cos2T      •  •  •       a"co8iY 

(    (57) 


wt=          Oinr+6;"sin2T+  •  •  •  +OintY, 
where 


__   _>  „_ 

3^.'  a>  - 

1        /  O  7Y«> 

«> 


(58) 


204.  Numerical  Example  2.  —  As  a  first  example  of  a  symmetrical 
periodic  orbit  we  consider  three  finite  bodies  revolving  in  circles  according 
to  the  straight-line  solution  of  Lagrange.  We  suppose  that  the  mass  M  , 
about  which  the  particle  revolves,  is  between  Mt  and  Ml  .  Choosing  M  for 
tin-  unit  of  mass,  we  select  M,  =  10,  Mt  =  5.  The  unit  of  distance  is  MA/,; 
and  it  follows  from  the  solution  of  the  quintic  equation  of  Lagrange*  that 
the  distance  M,M  is  /?,  =  0.77172  ....  The  unit  of  time  is  selected  so 
that  N  =  I,  and  the  period  of  the  solution  is  assigned  so  that  v  =  5;  whence 


•See  Moulton,  Celettial  Mtctumic*  (second  edition),  p.  312. 


416  PERIODIC    ORBITS. 

The  differential  equations  of  relative  motion  of  the  particle  are 


where  a  is  given  by  the  relation  i?a3  =  k2.     The  constant  A;2  is  determined  by 

i   /,,  .      M8    \  „. 

R         ^l*K' 


whence 

A;2  =  0.23763,         A;2^^  2.  37630,        ^^=2.58518. 


o=  1.05914/*,        •§-  =  1.37222  M. 

/Vj 

The  diJGFerential  equations  of  motion  become 


~  p         +JU+  P  =  (2-48074+7.44222 

-  (1.15938  cos  w?+  1.93230  cos3w)pV 

+  (4.23765+9.41700  cos2w;+16.47975cos4w)pV+ 


(60) 


+  (0.38646  sinw+1.93230  sin3w)pV 
-  (4.70850  sin2w+ 16.47975  sin  4w)pV+-  •  • 
The  periodic  solution  of  equations  (60)  is 
P  =  1  -  § M-  (0.27136+0.96615  cosr+4.96148  cos2r)M2 

+  (0.62584- 19. 13740  cos  r-2.47963  cos2r+0.40256  cos3r)M3+ ••  • , 

(61) 
W  =  T+  (1.93230  sin  r+ 6.82204  sin  2  TV 

+  (41. 10885  sinT+10.74793sin2r-0.48307sin3TVH 


I'AKTICLK    AI1KACTED    BY    It    Sl'HI.i 


417 


On  substituting  the  value  n  =  0.2,  the  orbit  represented  by  e<iuations 
«il)  is  shown  in  Fi^.  12.  The  points  which  are  numbered  1,  2,  ....  J  ! 
represent  positions  of  the  particle  in  the  periodic  orbit  at  intervals  of 
r  =  */12.  The  corresponding  positions  in  the  comparison  circle  are  indi- 
cated by  the  numbers  1',  2',  .  .  .  ,  24'. 


8    7 


•M 


18 


Fio.  12. 


205.  Numerical  Example  3.  —  For  a  second  example  of  a  symmetrical 
l>eriodic  orbit  we  use  the  configuration  (B),  §201,  of  nine  finite  bodies,  the 
numerical  values  l>eing  those  given.  The  unit  of  time  is  selected  so  that 
.V  =  1,  and  the  period  of  the  solution  is  assigned  so  that  V'  =  5,  whence  M  =  0.2. 

The  differential  equations  [corresponding  to  equations  (50)1  of  relative 
motion  of  the  particle  are 


'+  ?  -  2  '«  '"[* 


. 
/t, 


,'-  ir,)+35cos4(ic-  H'( 


P  j~ 


418  PERIODIC   ORBITS. 

On  taking  account  of  the  relations  (44)  ,  and  choosing  the  unit  of  distance 
so  that  -4  =  1,  the  preceding  set  of  equations  takes  the  form 


~1 

-J 


ISM"  ,  fas/a 
~~ 


From  §201  we  have  the  following  values: 
K  =  2,        M'=l,        M"  = 


Since  AT  =  1,  the  last  equation  gives  A;2  =  0.60994.     It  follows  that 

P  M"  /  n  \  2 

~-  =0.62920,         a2  =  2.10300M2,         (|?)  =0.52575  /A 

On  substituting  these  numerical  values,  the  differential  equations  become 
~  P®  +»y+  ~*  =  2-47828  pM2+[3.63039+8.32914cos4io]pV+  •  •  •  , 


The  periodic  solution  of  these  differential  equations  is 
P=  l-M-0.27054M*+1.70909M'-3.43127M4-0.8329lM4cos4r+ 


On  substituting  the  value  M  =  0.2,  the  final  result  is  found  to  be 

P  =  0.86403-0.00133cos4r+  •  •  •  , 
w  =  r+0.00150sin4T+ 

The  orbit  has  four  axes  of  symmetry,  namely,  the  lines  connecting  the 
central  body  with  the  others.  It  differs  from  the  orbits  of  the  other  numer- 
ical examples  in  one  respect  —  that  is,  it  lies  entirely  inside  the  comparison 
circle  (see  §200).  In  terms  of  p  the  radius  of  the  comparison  circle  is  about 
0.8855.  Fig.  13  is  not  drawn  to  scale,  but  the  characteristic  properties  of 
the  orbit,  which  are  readily  seen  from  the  numerical  values  of  p  and  w,  are 
exaggerated  to  make  them  apparent  in  a  small  drawing.  The  inner  circle 
is  drawn  merely  to  indicate  the  direction  of  the  deviation  of  the  orbit  from 
a  circle. 


PARTICLE    ATTRACTED    BY    H    SIMIKKi:-. 


419 


206.  The  Undisturbed  Orbit  Must  be  Circular.  In  tin-  proofs  of  the 
exigence  of  periodic  orbits  (§§198-201)  it  was  assumed  that  the  undisturbed 
orl>it  is  circular.  It  remains  to  be  shown  that  this  assumption  is  necessary. 
The  proof  will  be  made  for  the  case  of  symmetrical  orbits  [equations  (50)] 
and  is  applicable  also  to  the  orbits  of  §198.  The  undisturbed  orbit  is  given 
by  the  solution  of  equations  '>()>  when  ^  =  0.  For  M  =  0,  the  equations  are 
the  equations  of  motion  of  a  particle  subject  to  the  attraction  of  a  central 
force  varying  inversely  as  the  square  of  the  distance.  The  undisturbed  orbit 
is  therefore  a  conic ;  and,  since  we 
are  concerned  only  with  closed 
orbits,  must  bean  ellipse.  Since 
the  period  in  r  is  2*,  the  major 
semi-axis  of  the  ellipse  must  be 
unity  (in  p).  The  eccentricity, 
which  will  be  denoted  by  e,  is, 
however,  arbitrary;  that  is,  for 
M  =  0  the  differential  equations 
admit  an  infinite  number*  of 
symmetrical  periodic  solutions. 
Starting  now  with  an  ellipse  for 
the  undisturbed  orbit,  it  will  be 
shown  that  the  eccentricity  must 
be  zero  in  order  to  fulfill  the 
conditions  of  periodicity. 

For  /u=0  the  solution  of  equa- 
tions (50)  representing  an  elliptic  orbit  of  eccentricity  e  is 


*. 

FIGURE  13. 


p=  1  —  ecosE, 


C(  »  /'.'         . 

i-ecosE-> 


- 
—ecosE 


where  E  is  defined  by  the  relation 
r  =  0  arc 

P  =  l-«,        P'  =  0, 


E—em\E.     The  initial  conditions  for 


w' 


%  i-,'7 
(T-e)* 


Consider  the  solution  for  values  of  n  different  from  zero,  but  sufficiently 
small,  and  let  the  initial  conditions  be 


v'l -?.i...        x'l-Ce+c)* 

/; — ^s  ~rP4—  /.   •  _\iW. — ri  i  -\li      M- 


If  a  and  e  can  be  determined  in  terms  of  n,  vanishing  with  /*,  so  that  the 
conditions  (52)  are  satisfied,  then  the  solution  will  he  periodic  with  the 

•The  general  CMC  when  the  differential  equation*  (for  M-0)  admit  a  periodic  solution  containing  an 
arbitrary  parameter  has  been  mentioned  by  Poincart,  toe.  «<.,  vol.  1,  p.  84. 


420  PERIODIC   ORBITS. 

period  2ir.  All  terms  of  the  solution  which  are  independent  of  ju*  may  be 
obtained  from  the  two-body  problem  by  making  the  substitution  w  =  u  —  /rr. 
These  terms  are  given  in  finite  form  by  the  expressions 


u  =  are  cos  = 

M  —  (je+e)cosE/ 

where  E  is  defined  by  the  relation 


),/t  =E-(e+e)smE. 

On  returning  to  the  variable  w,  writing  the  terms  in  a  and  e  as  power 
series  by  Taylor's  expansion,  and  applying  the  conditions  (52),  we  obtain 
the  equations 


It  follows  from  the  known  properties  of  the  series  that  there  are  no  terms  in 
e  alone,  and  there  are  no  terms  involving  /i  to  the  first  degree  except  the 
term  —  PTTJU.  Hence  the  second  of  equations  (63)  can  be  solved  for  a  as  a 
power  series  in  e  and  /*  in  which  ^  is  contained  as  a  factor;  the  result  is 


a  =  MT  —  T 


When  this  value  of  a  is  substituted  in  the  first  equation,  a  factor  /j.  can  be 
divided  out,  leaving 


This  equation  can  be  solved  for  e  as  a  power  series  in  n,  which  vanishes  with  n, 
if  and  only  ifH  =  Q.  Since  only  those  solutions  are  under  consideration  which 
are  the  analytic  continuations  with  respect  to  /*  of  those  for  n  =  Q,  the  con- 
dition e  =  0  must  be  imposed.  The  condition  e  =  Q  means  that  the  undisturbed 
orbit  must  be  circular. 

207.  More  General  Types  of  Motion  for  the  Finite  Bodies.  —  This  sec- 
tion contains  some  remarks  upon  possible  extensions  of  the  analysis  which 
will  permit  applications  to  practical  problems  of  celestial  mechanics,  and  is 
followed  by  an  illustrative  example. 

The  particular  problems  treated  in  the  preceding  articles  have  no  appli- 
cation in  nature  because  the  configurations  assumed  for  the  finite  bodies  do 
not  exist.  But  a  glance  at  the  details  shows  that  these  configurations  are 
not  essential  to  the  proofs.  The  possible  generalizations  of  the  motion  of  the 
finite  bodies  can  be  made  in  three  ways: 


PAKTK'KE    ATTKACTEU    BY    «    SPHKIiKS.  421 

(1)  In  §198  the  existence  proof  depends  only  upon  certain  terms  of 
the  disturbing  function  which  are  due  to  the  body  Ml.     If  A/,  retains  the 
motion  there  prescribed,  we  may  add  other  bodies  to  the  fixed  configuration 
in  the  rotating  plane  provided  the  operations  with  the  power  series  are  valid. 
This  merely  increases  the  number  of  terms  in  the  second  members  of  the  equa- 
tions of  motion ;  the  existence  proof  and  method  of  construction  are  unchanged . 

(2)  In  the  examples  treated  the  finite  bodies  form  a  fixed  configuration 
in  a  plane  rotating  with  constant  angular  velocity.    This  is  not  necessary 
for  the  type  of  analysis  used.    If  M,  moves  in  a  circle  with  uniform  angular 
velocity,  the  other  bodies  can  have  any  periodic  motion,  provided  always 
that  the  convergence  conditions  hold.     In  this  case  the  differential  equations 
of  motion  of  the  particle  involve  r  explicitly  and  are  periodic  in  r.    Two 
points  of  difference  occur  in  the  analysis:  (a)  Suppose  the  period  in  r  of 
the  differential  equations  is  T;  then  the  assigned  period  of  the  motion  of 
the  particle  must  be  a  multiple  of  T.     (b)  The  differential  equations  do 
not  admit  the  integral  of  Jacobi,  and  hence  no  use  can  be  made  of  this  in 
the  existence  proof.    This  is  equivalent  to  saying  that  at  r  =  0  we  can  not 
assume  u;  =  0,  but  must  determine  the  initial  longitude  of  the  particle  by 
the  conditions  of  periodicity.    The  method  of  determining  the  constants  of 
integration  in  the  construction  of  the  solutions  is  explained  in  a  paper  in  the 
Transactions  of  the  American  Mathematical  Society,  vol.  8  (1907),  pp.  177-181. 

(3)  A  further  generalization  of  the  motion  of  the  finite  bodies  is  pos- 
sible by  permitting  M,  to  move  in  a  path  which  is  not  circular.    It  is  possible 
to  show  that  the  analysis  can  be  used  if  the  motion  of  A/,  is  subject  only  to 
the  mild  restrictions  that  the  expression  for  the  radius  vector  shall  contain 
only  cosines  of  multiples  of  T  while  that  for  the  longitude  shall  contain 
only  sines.    The  case  when  the  orbit  of  Af ,  is  an  ellipse  is  treated  in  the 
article  referred  to  above.    For  this  generalized  motion  of  the  finite  bodies 
there  may  exist  symmetrical  orbits  of  the  particle.    In  equations  (50)  the 
first  contains  only  cosines  of  multiples  of  w,  and  the  second  only  sines  of 
multiples  of  u;.    The  periodic  orbit  of  the  particle  may  be  symmetrical  if 
the  first  equation  contains  also  sines  of  multiples  of  w  multiplied  by  odd 
functions  of  r,  and  cosines  of  multiples  of  to  multiplied  by  even  functions 
of  T,  and  the  second  contains  cosines  of  multiples  of  w  multiplied  by  odd 
functions  of  r,  and  sines  of  multiples  of  to  multiplied  by  even  functions  of  T. 

From  these  remarks  it  is  apparent  that  the  treatment  can  be  made 
sufficiently  general  to  permit  applications  in  the  problems  presented  by  the 
motions  of  the  solar  system.  For  example,  suppose  P  is  a  satellite  of  one  of 
the  planets  M,  and  that  A/,  is  the  sun .  This  implies  that  the  disturbing  effects 
of  the  satellite  upon  the  other  bodies  are  neglected,  since  we  assume  that  its 
mass  is  infinitesimal.  The  conditions  upon  the  motion  of  Af,  are  fulfilled 
if  we  neglect  the  perturbations  of  the  other  planets  upon  M;  that  is,  if  we 
suppose  the  orbit  of  A/,  relative  to  M  is  an  ellipse.  If  we  neglect  the  incli- 
nations of  the  orbits  of  the  other  planets,  and  suppose  that  their  motion  is 


422  PERIODIC    ORBITS. 

periodic  (that  is,  we  assign  a  periodic  motion  which  is  approximately  correct), 
it  is  possible  by  the  methods  given  to  treat  the  periodic  motion  of  the  satel- 
lite in  the  plane  of  the  planetary  orbit,  when  subject  to  the  attraction  of  the 
sun  and  all  the  planets.  The  following  numerical  example  is  a  simple 
illustration  of  the  general  idea. 

208.  Numerical  Example  4.  —  The  mass  of  M  is  taken  as  the  unit  of 
mass  and  Mlt  of  mass  10,  is  supposed  to  revolve  about  M  in  a  circle  of  unit 
radius  with  uniform  angular  velocity  N.  A  third  mass,  Mt  =  M  =  1,  is 
supposed  to  revolve  about  Mt  in  a  circle  of  radius  At  with  uniform  angular 
velocity  Nt.  The  unit  of  time  is  chosen  so  that  N=l,  and  the  period  of 
the  motion  of  the  particle  is  assigned  so  that  v  =  5,  whence 

N 
H  =  m=  —  =0.2. 

v 

With  reference  to  M  as  origin  and  an  axis  passing  always  through  Ml,  the 
coordinates  of  Mlt  Mt,  and  P  are,  respectively  (1,0),  (Rt,  Ws),  and  (r,  w). 
The  differential  equations  of  relative  motion  of  the  particle  [corresponding 
to  equations  (50)]  are 


16 


-  Wt)  +  f  |-P{sin(«;-  WJ  +  5sin3  (w-  Wt)  J 


(64) 


The  constant  A;2  is  given  by  the  relation     N*  =  k*(M-}-Ml),  whence 

#  =  0.09091,         A;W!  =  0.90909,         k*Mt  =  0.09091. 
From  the  relation  v*a3  =  k*M,  it  follows  that 


The  angular  velocity  of  Mt  about  Ml  will  be  selected  so  that  its  period  with 
respect  to  the  rotating  axis  MMl  is  one-half  the  period  assigned  for  P. 
Hence 

Nt-N  =  2v,     or    #,=  11. 

The  radius,  At  ,  of  the  circular  orbit  of  Mt  with  respect  to  Ml  is  determined 
by  the  relation  N$A*t  =  ^(.M^+M,)  .  whence  A9  =  1  .01200ju.  On  assuming  that 


PARTICLE   ATTRACTED    BY    U   SPHERES.  42.3 

at  T  =  0the  finite  bodies  are  in  conjunction  in  the  order  M,  A/,,  A/,,  the  coordi- 
nates (/(",,  M',)  of  A/f  with  respect  to  M  are  given  by  the  expressions 


coslf  = 

^          2ft 


On  substituting  the  values  of  the  constants  and  the  coordinates  ft,  M  , 
in  equations  (64),  we  obtain  for  the  numerical  differential  equations  of 
relative  motion 


-*(4r  +M  )'+-,  =(0.50000  +  1.50000  cos  2u>)pM« 


—  vw.wvrw ~r~  * .tjwwv/vro ttw mi* 

p 

+  (0.86523  cos  w+ 1.44205  cos  3u>)pV 
+  (0.33273 + 0.73940  cos  2u> + 1 .29395  cos  4u?)pV 
+  (0.27600sin2rsin2t0-0. 13800  cos2T-0.41400co82rco82u>)pM' 
+  (0.10474+0.17456  cos4T+0.31422cos2u>+0.10471  cos  4rcos2u> 
-0.55864  sin  4r  sin  2u>)pM4 

+  (0.07960sin2Tsinu>-0.31840cos2rcost0-0.53068cos2TC083u> 
+0.39801  sin2rsin3u>)pV+  •  •  •  ,  (65) 

+2^(^  +M)  =  -(1.50000sin2u;)pMt 

-  (0.28841  sin  u>+ 1.44205  sin  3u>)pV 

-  (0.36970sin2u>+ 1 .29395  sin  4u;)pV4 

+  (0.27600  sin  2r  cos  2w> +0.4 1400  cos  2r  sin  2u>)pM* 
-(0.31422  sin  2u>+0.10741  cos  4r  sin  2ic+0.55864  sin  4rcos2u;)pM4 
+  (0.02653  sin2TCOSU>+0.10613cos2rsint/;+0.39801sin2TCos3u> 
+0.53068cos2Tsin3uj)pV+  •  •  •  • 

The  right  member  of  the  first  equation  contains  only,  (1)  cosines  of 
multiples  of  w,  (2)  cosines  of  multiples  of  w  multiplied  by  cosines  of  multiples 
of  T,  and  (3)  sines  of  multiples  of  w  multiplied  by  sines  of  multiples  of  T. 
The  first  equation  is  then  unchanged  if  we  replace  w  by  -  w,  and  T  by  -  T. 
The  right  member  of  the  second  equation  contains  only,  (1)  sines  of  mul- 
tiples of  w,  (2)  sines  of  multiples  of  w  multiplied  by  cosines  of  multiples  of  T, 
and  (3)  cosines  of  multiples  of  w  multiplied  by  sines  of  multiples  of  T.  Hence 
the  second  equation  is  also  unchanged  if  we  replace  w  by  —  w  and  T  by  —  T. 
Now  let  us  suppose  that 

is  a  solution  of  equations  (65)  satisfying  the  conditions  p'(0)  =u;(0)  =  0.  It 
follows  from  the  form  of  the  differential  equations  that  ^,  is  an  even  function, 
and  ^t  is  an  odd  function  of  T.  When  r  =  0  the  finite  bodies  are  in  conjunction 
in  the  order  M,  A/,,  A/,.  Therefore,  if  the  particle  P  crosses  the  line  A/ A/, 
orthogonally  when  the  finite  bodies  are  in  conjunction  in  the  order  M,  A/, ,  A/t,  the 
<>rl>H  In  the  rotating  plane  is  symmetrical  with  respect  to  this  line  and  this  epoch. 


424 


PERIODIC   ORBITS. 


On  constructing  the  solution  of  equations  (65)  by  the  formulas  (58),  we  get 

P  =  l-  .66667^+  (0.38889+0.72102  cosr-cos2r)M2 

+  (-0.02616+2.09168  cos  r- 0.45400  cos  2r- 0.30043  cos  3r 

+0.03450  cos  4rV+  •  •  •  . 

((50) 
w  =  T+  (- 1.44204  sin  r+ 1.37500  sin  2r)Ml 

+  (-6.29838  sinT+2.12066sin2T+0.36051sin3T 
-0.03881  sin  4rV+ 

The  orbit  represented  by  equations  (66)  is  shown  in  Fig.  14.  The 
points  which  are  num- 
bered 1, 2, .  .  .  , 8  rep- 
resent the  positions  of 
the  particle  in  the  peri- 
odic orbit  at  intervals 
of  T  =  7T/4 .  The  corre- 
sponding positions  in 
the  comparison  circle 
are  indicated  by  the 
numbers  1',  2',...,  8'. 

With  reference  to 
the  differential  equa- 
tions (65)  we  can  make 
the  same  statement 
concerning  the  unique- 
ness of  the  solution 
that  was  made  in  §  198. 
These  equations  were 
written  on  the  assump- 
tion that,  at  T  =  0,  the  finite  bodies  are  in  conjunction  in  the  order  M,  Ml ,  M, . 
Without  this  assumption  the  expressions  for  Rt  and  Wt  contain  a  parameter 
indicating  the  position  of  Mt  in  its  orbit  at  the  origin  of  time.  With 
reference  to  the  physical  problem,  therefore,  we  can  not  affirm  in  this  case 
that,  for  a  preassigned  period,  there  exists  one  and  only  one  direct  periodic 
orbit.  It  is  necessary  here  to  add  a  condition  on  the  form  of  the  configu- 
ration of  the  finite  bodies  and  particle  at  the  origin  of  time.  For  example, 
we  might  obtain  another  symmetrical  periodic  orbit  having  the  preassigned 
period  if  the  particle  crosses  the  line  M,  Ml  when  the  finite  bodies  are  in 
conjunction  in  the  order  M,  Ms,  Ml;  and  we  might  have  still  other  orbits 
with  the  preassigned  period  if  P  crosses  this  line  when  the  finite  bodies  are 
not  in  conjunction. 


Fro.  14. 


CHAPTER  XIV. 

CERTAIN  PERIODIC  ORBITS  OF  /•  FINITE  BODIES 

REVOLVING  ABOUT  A  RELATIVELY  LARGE 

CENTRAL  MASS. 


BY  FRANK  LOXLEY  GRIFFIN. 

209.  The  Problem. — For  a  given  system  of  k  finite  bodies,  moving  in  a 
given  plane  relative  to  another  given  body,  there  is  a  4fc-fold  infinitude  of 
possible  orbits — the  variations  which  the  configuration  of  the  system  under- 
goes and  its  orientation  in  the  plane  being  determined  jointly  by  the  mutual 
attractions  of  the  bodies  according  to  the  Newtonian  law,  and  by  the  values 
at  any  instant  of  the  2k  relative  coordinates  and  their  first  derivatives  with 
respect  to  the  time.  The  differential  equations  admit  no  algebraic  or  uni- 
form transcendental  integrals,*  aside  from  the  two  fundamental  integrals  of 
energy  and  areas,  even  when  the  masses  of  all  the  bodies  except  one  are 
very  small;  nevertheless,  by  restricting  the  initial  values  of  the  coordinates 
and  their  derivatives,  in  a  manner  to  be  shown  below,  it  is  possible  to  find 
an  extensive  class  of  periodic  solutions. 

In  fact,  for  arbitrary  values  of  the  masses  (save  that  one  of  them,  M , 
shall  be  large  in  comparison  with  the  others,  M,,  Mt,  .  .  .  ,  A/»),f  there 
exists  a  k-fold  infinitude  of  distinct  periodic  orbits  of  the  system,  having  an 
arbitrarily  preassigned  period  T.  In  these  orbits  (which,  for  small  finite 
values  of  Mlt  Mt,  .  .  .  ,  Mtl  depart  but  little  from  a  set  of  concentric 
circles  about  the  planet)  the  k  satellites  come  periodically  into  a  "symmet- 
rical conjunction,"  that  is,  they  are  all  momentarily  in  one  straight  line  with 
the  planet  and  moving  at  right  angles  to  that  line.  These  conjunctions  may, 
or  may  not,  always  occur  at  the  same  absolute  longitude;  in  the  latter  case 
the  motion  is  periodic  with  reference  to  a  uniformly  rotating  line. 

Besides  the  demonstration  of  the  existence  of  such  periodic  orbits,  this 
chapter  contains :  A  method  of  constructing  the  solutions  without  integration, 
a  single  application  of  that  process  having  provided  formulas  which  reduce 
the  problem  to  one  of  algebraic  computation ;  a  numerical  application  to  the 
case  of  Jupiter's  satellites  I,  II,  and  III;  a  proof  of  the  non-existence  of  cer- 
tain other  types  of  orbits;  and  a  brief  consideration  of  some  related  questions. 

•See  memoirs  by  Bruns  and  Poincare,  in  Ada  MaAematica,  vols.  1 1  and  13. 

tin  other  words,  the  distribution  of  masses  is  such  as  is  presented  by  the  sun  and  any  number  of  planets, 
or  by  a  planet  and  any  number  of  satellites.  For  convenience,  in  what  follows,  a  single  expression,  planet 
and  satellites,  will  be  used  with  the  understanding  that  it  covers  also  sun  and  planets. 

426 


426  PERIODIC   ORBITS. 

The  problem  may  be  formulated  thus :  Let  quantities  ju,  ft ,  .  .  .  ,  ft  be 
defined  by 

M^  =  Mi  (i  =  i,  ...,*),          (1) 

where  one  of  the  j8's  is  to  be  selected  arbitrarily.  Let  a  system  of  positive 
or  negative  integers  without  common  divisor,  pt(i  =  l,  .  .  ,  fc  — 1),  and  a 
number  <7t5^0  be  selected  arbitrarily,  save  for  the  restriction  mentioned 
below,  and  let  v ,  nt ,  and  a,  be  defined  by 

_  2?r  _  nt  _  nt—nt         ..  ,      .         (ey\ 

"~"    ~  »-!•  ••*- 


(t  =  l,...,*),         (3) 

where  K*  denotes  the  gravitational  constant,  and  where,  of  the  three  values 
of  a,  satisfying  (3),  that  one  is  to  be  selected  which  is  real.  Also  let  the 
notation  be  so  selected  that  al}  .  .  .  ,  at  are  in  ascending  order  of  magnitude, 
the  PI  being  so  selected  that  no  two  of  the  ck  are  equal  and  no  n(  vanishes. 

If  M  were  zero — that  is,  if  the  satellites  were  "infinitesimal" — possible 
orbits  would  be  circles  about  the  planet  with  c^,  .  .  .  ,  at  as  radii;  from  (3) 
it  follows  that  the  angular  velocities  would  be  r^,  .  .  .  ,  nt.  It  is  quite 
immaterial  whether  any  of  the  nt  are  negative;  the  results  obtained  hold 
irrespective  of  retrograde  motion  of  some  of  the  bodies.  The  configuration 
of  the  infinitesimal  system  would  undergo  periodic  variations  with  the 
period  T;  for,  it  foUows  from  (2)  that 

27T        =   T} 

or  each  synodic  period  is  a  sub-multiple  of  T.  This  condition  being  satisfied,* 
the  motion  of  the  infinitesimal  system  would  be  periodic  with  respect  to  a 
line  through  M,  rotating  with  uniform  angular  velocity — that  of  Mt,  or, 
indeed,  that  of  any  other  bodyf  M , — though  whether  or  not  the  system 
ever  returns  to  the  same  position  in  space  depends  upon  whether  qt  is  rational 
or  irrational. 

In  describing  the  orbits  mentioned,  the  infinitesimal  satellites  would  be 
subject  to  certain  initial  conditions,  the  2k  coordinates  and  their  derivatives 
with  respect  to  the  time  having  at  the  instant  t  =  t0  certain  values,  say 
Co(t=l,  .  .  .  ,k;j=l,  .  .  .  ,4);  but  if  the  k  finite  satellites  are  subjected 
to  these  same  initial  conditions,  their  mutual  disturbances  in  general  destroy 
periodicity.  The  first  problem  is,  then,  to  determine  what,  if  any,  incre- 
ments Acw  can  be  given  to  the  former  initial  values  di  to  preserve  the 
periodicity  when  all  the  satellites  are  finite. 

*Poincar6,  treating  three  satellites  (Mtthodes  Nouvelles  de  la  Mtcanique  C&este,  vol.  1,  pp.  154-6),  states 
the  condition  thus:  Integers  o,  ft,  y,  mutually  prime,  exist  such  that  0+^+7=0  and  anl+ftn,+yn,=0. 
Evidently,  in  the  case  of  three  satellites,  this  condition  ia  equivalent  to  (2),  since  (n,—  n,)//J  =  (n,—  n,)/(  —  o); 
but  for  a  greater  number  it  is  not  so.  Thus,  if  n,=7,  n,  =  5,  n,=3  v^7  n,  =  V2~t  the  integers  5,  —7,  —1,  3 
satisfy  a  condition  similar  to  Poincar6's,  but  periodicity  is  impossible.  For  the  general  case  a  re-formulation 
such  as  (2)  is  necessary. 

fThe  commensurability  of  n«— nt(i  =  l,  .  .  .  ,  k  —  1)  evidently  involves  that  of  n»— n/  (f  =  l,  .  .  .  , 
j  —  l,j+l,  .  .  .  ,  fc).  For  from  nt—nt  =  ptv  and  n;— nt  =  p;K  it  follows  that  ni—nl"(pi—p1)». 


PROBLEM    OF   k   SATELLITES.  427 

210.  The  Differential  Equations. — Let  the  common  plane  of  relative 
motion  of  the  k  bodies  be  selected  as  the  £  //-plane,  the  origin  being  at  M, 
:uid  ME  and  M II  being  rectangular  axes  which  rotate  in  the  plane  with  the 
uniform  angular  velocity  N.  Let  the  coordinates  of  A/,  referred  to  these 
axes  be  £,  and  T;,;  then  the  differential  equations  of  motion  are 


(«)         *3-  •^^-N>ll+t(M+Mt)*--Z>SMl(«S>    .*)=0, 

(4) 

(6)         5J  +  2N  ^  -  A-'ib+^V+M ,)  J' 
ar  at  77 

where 


mid  -'  means  2'!*  QVi)-  Except  in  proving  a  certain  symmetry  theorem, 
t  hese  coordinates  are  less  convenient  than  polar  coordinates  referred  to  rotat- 
ing reference  lines.  Besides  (1),  (2),  and  (3),  let  the  following  definitions 
be  made: 

itj  =  a  Jo., ,        vqt  =  n,  (»-!,...,*), 

(5) 


where  the  X,  are  arbitrary  constants,  later  to  be  taken  as  the  longitudes  of 
the  Mi  at  the  origin  of  time  for  n  =  0. 

Let  polar  coordinates  be  introduced  by  the  equations 

£,  =  r,cosu,,        ij,  =  r,smui,        rt  =  atxl,        ut  =  w{-\-pir-\-\, ,          (6) 

and  N  be  taken  equal  to  n»,  so  that  w,  is  the  longitude  of  3f,  referred  to  a 
line  rotating  with  uniform  speed  n, .     The  differential  equations  become 


(7) 


(6) 


-».)          -  =0, 


where  a]a]J  =  a\^i+a]xLJ-2ataixtxJcos(4>Jl+w)-wl),  and  where  the  accents  on 
the  variables  indicate  derivatives  with  respect  to  T. 

211.  Symmetry  Theorem.  —  //  a  symmetrical  conjunction  occurs  at  any 
instant  t  =  t,,  then  the  orbit  of  each  satellite  before  and  after  the  conjunction  is 
symmetrical,  both  with  regard  to  geometric  equality  of  figures  and  witii  regard 
to  intervals  of  lime.  A  proof  will  be  given  only  for  the  case  <,  =  0,  which  cloc- 
not  limit  the  generality  since  any  other  case  is  reduced  to  this  one  by  the 
substitution  <  = 


428  PERIODIC   ORBITS. 

The  differential  equations  (4)  are  invariant  under  the  substitution 

?<  =  &,  if  =-17.,  l=-t.  (8) 

Consequently,  every  solution  of  (4)  is  transformed  by  (8)  into  some  solution 
of  (4).     Moreover,  the  initial  conditions 

&  =  *,         *  =  0,         §'=0,         4j*=b.  (9) 

are  transformed  into 

?<  =  «.,        *  =  0,        ^=0,         g'=6,.  (9') 

Therefore,  that  solution  of  (4)  which  satisfies  the  initial  conditions  (9)  is 
transformed  by  (8)  into  itself.     Hence,  if  that  solution  is 

«,  =  *,(*),        *  =  *,(*)  (t  =  l,  ...,*),         (10) 

then 

*«(0  =*«(!)  =*i(-0,       *«(0  =  -*.(«)  =  -*,(-«), 
whence  also 


It  will  be  noted  that  the  proof  holds,  whatever  the  value  of  N.  It  is 
also  geometrically  evident  that  the  symmetry,  if  present  at  all,  is  independent 
of  the  rate  of  rotation  of  the  reference  line. 

212.  Conditions  for  Periodic  Solutions.  —  Since  the  differential  equations 
(7)  are  unchanged  if  r  is  replaced  by  T+2mr,  or  t  by  t+nT  (n  being  an 
integer),  it  follows  that  if 


(t  =  l,  ..>.,*)      (11) 
is  a  solution,  then  so  is 

xt  =  xt(T+2nir),        wl  =  w,(T+2mr).  (12) 

These  two  will  be  the  same  solution  if  the  coordinates  and  their  derivatives 
have  the  same  values  at  r  =  r0;  that  is,  if 


x't(rl)+2mr}=x't(T0),         wfa^nir)  =w'i(r0).  ) 


If  these  conditions  are  satisfied,  then,  for  all  values  of  T, 

r)  =XI(T),         wt 


that  is,  (13)  are  sufficient  conditions  for  the  periodicity  of  the  solutions.    That 
they  are  also  necessary  is  obvious,  if  the  period  is  to  be  2mr. 


I'HOBI-KM    OK    k    S.MKI.1.IIE8. 

Special  case.  —  In  the  case  of  a  symmetrical  conjunction  at  r  =  0,  other 
sufficient  conditions  can  he  formulated.  For,  if  x'(0)  =u?,(0)  =0  (t-l,  .  .  .  ,  fc), 
and  if  every  X,  is  a  multiple  of  T,  it  follows  from  the  symmetry  theorem  that 

-T),  1 

-,T).  J 

But,  by  equations  (13),  if  T,  is  put  equal  to  -T,  the  conditions  for  period- 
icity of  X,(T)  and  U»,(T)  are 


(a)  *,(*•)=*,(-*•).         (c)     wt(ir)  =•«>,(  -T),  J 

(b)  *;(*)=  *;(-*),     (d)  «,;(*)  =*;(-*).         } 

Of  these  conditions  (a)  and  (d)  are  satisfied  by  virtue  of  (14),  while  (6)  and 
(c)  are  also  satisfied  if  z'(n-)  =  U?,(T)  =0.  It  may  then  be  stated  that  sufficient 
conditions  for  the  periodicity  of  x,  and  w,  (with  period  in  t  equal  to  T)  are 

(a)  *;«>)=0,        tr,(0)  =  0,        X,=OorT,  J 

(6)  *;«=<),         «>,(ir)=0.  J 

Moreover,  after  conditions  (16a)  have  been  imposed,  conditions  (6)  are 
necessary  as  well  as  sufficient. 

213.  Nature  of  the  Periodicity  Conditions.  —  For  M  =  0  the  differential 
ec  (nations  (7)  admit  the  solution  with  period  2r  (or  T  in  0 


giving  the  circular  orbits  r,=a,  ,  w,=X,+p,T,  in  which  at  T-0,  x,-l,  if  =tp,=t01-0. 
If  these  initial  values  are  given  increments  Ac,j(t=l,  .  .  .  ,  k;  j=l,  2,  3,  4), 
then  the  solutions  of  the  differential  equations  (7)  for  M^O  are  developable 
as  power  series  in  n  and  the  Ac,,  ,  which  converge  throughout  a  preassigned  in- 
terval of  T  for  sufficiently  small  values  of  those  parameters.*  Such  solutions 
are  in  general  non-periodic;  in  fact,  the  periodicity  conditions  (13)  or  (16) 
impose  the  condition  that  4Jk  power  series  in  these  4fc+l  parameters  shall 
vanish.  In  the  cases  to  be  considered  these  4A;  equations  will  determine 
the  Ac,,  as  unique  functions  of  n,  holomorphic  in  the  vicinity  of  n  =  Q  and 
vanishing  with  n;  so  that,  for  sufficiently  small  values  of  M,  there  exist  initial 
conditions  (depending  upon  T,  qu  the  p,  ,  M,  and  the  ft)  such  that  the  orbits 
described  are  periodic  with  the  required  period. 

Evidently  for  smaller  and  smaller  values  of  /u,  smaller  and  smaller  devi- 
ations from  the  initial  conditions  of  undisturbed  motion  are  sufficient  in 
order  to  get  periodic  orbits.  These  orbits  for  ^^0  may  be  said  to  "grow  out 
of"  the  undisturbed  circular  orbits  as  M  grows  from  zero.  Of  course,  for 
any  given  masses,  n  and  the  0,  being  fixed,  the  possible  orbits  of  this  sort  can 

•See  fJU-16. 


430  PERIODIC    OUBITS. 

vary  only  with  T,  qt,  or  the  p( ;  but  to  a  range  of  values  of  /j,  there  corresponds 
a  class  of  orbits. 

In  what  follows  it  will  be  inquired  whether  the  conditions  for  periodicity 
can  be  satisfied  by  such  values  of  the  Ac,,  as  to  prove  the  existence  of  a 
class  of  periodic  orbits  of  each  of  the  following  types : 

Type  I.     The  finite  system  has  a  symmetrical  conjunction. 

Type  II.  The  infinitesimal  system  has  a  symmetrical  conjunction,  but 
the  finite  system  has  none. 

Type  III.  Neither  system  has  a  symmetrical  conjunction. 

214.  Integration  of  the  Differential  Equations  as  Power  Series  in 
Parameters. — It  will  be  necessary  to  obtain  the  first  few  terms  of  the  devel- 
opments mentioned  in  the  preceding  article.  Instead  of  increments  Ac,, 
to  the  initial  undisturbed  values  of  the  coordinates  it  will  be  more  convenient, 
in  finding  the  properties  of  the  solutions,  to  employ  parameters  An, ,  e, ,  w, ,  T,  , 
defined  as  follows.  At  T  =  0  let 


(17) 


the  vt  being  equal  to  ti,+gtr,  the  true  longitudes  from  a  fixed  reference  lino. 

It  is  evident  that  the  Acw  are  holomorphic  functions  of  the  An,,  e(,  w,, 
and  rt  for  sufficiently  small  values  of  the  latter  quantities.  Consequently, 
solutions  of  (7)  exist  also  as  power  series  in  the  new  parameters.  Further, 
since  the  real  positive  values  of  the  radicals  and  the  smallest  values  of  the 
inverse  cosines  are  to  be  taken  in  (17),  the  Ac,y  are  given  uniquely  in  terms 
of  the  An, ,  et ,  w< ,  and  T<  .  From  these  two  facts  it  follows  that,  if  the  latter 
quantities  can  be  determined  as  unique  power  series  in  n,  satisfying  the  con- 
ditions for  periodicity,  then  also  there  exist  for  the  Ac,,  unique  power  series 
in  ju,  satisfying  the  conditions.  Conversely,  while  the  Jacobian  of  the  Ac,, 
with  respect  to  the  new  parameters  is  zero  for  An,  =et  =  o>,  =  T,  =0,  yet,  in 
the  only  case  where  discussion  will  be  necessary  (viz.,  for  Ac(,=  Acf3  =  0, 
whence  u,  =T,  =  0),  the  solution  for  the  An,  and  et  in  terms  of  the  Ac,,  and 
Ac,  4  is  unique;  for  the  Jacobian  of  the  Ac(l  and  Ac,4  with  respect  to  An,  and 
e,  is  distinct  from  zero  for  An,  =  e,  =  0.  Hence,  in  this  case,  if  the  Ac,  t  and 
Ac,  4  exist  as  unique  series  in  /*,  satisfying  the  periodicity  conditions,  so  also 
must  the  An,  and  ef  exist  as  such  series. 

In  the  developments  of  the  coordinates  as  power  series  in  ju  and  the  new 
parameters  all  those  terms  independent  of  n  may  be  obtained,  together  with 
a  knowledge  of  theirjproperties,  in  the  following  simple  manner:  The  terms 


1'KOBLEM    OF  A'   SA1  KI.I.I  I  B8.  HI 

in  question  are  those  remaining  when  n  is  put  equal  to  zero,  and  are  there- 
fore the  solution  of  the  problem  of  A  infinitesimal  satellites  when  the  initial 
conditions  are  (17) — in  other  words,  the  solutions  of  k  two-body  problems. 
The  dynamical  meaning  of  the  new  parameters  is  then  evident.  The 
orbits  of  the  infinitesimal  system,  subjected  to  the  initial  conditions  (17),  are 
ellipses  in  which  the  mean  angular  motions,  major  semi-axes,  eccentricities, 
longitudes  of  pericenter,  and  times  of  pericenter  passage  are  respectively 

n,(l+AnO,         o,(l+AnO~l,         ft,         X,+u(,         -• 


If  in  the  development  of  x,  the  coefficient  of  A»f  cj  o>?  r?  be  denoted  by  x,iAM 
then,  by  applying  Taylor's  theorem  to  the  well-known  developments  of 
the  coordinates  in  elliptic  motion  as  power  series  in  the  eccentricity,*  it  is 
found  that  the  following  coefficients  of  first  and  second  degree  terms  do  not 
vanish : 


(18) 


,=28019,7-, 

u>t,nn  =  —  2q,coaq{r, 


From  simple  dynamical  considerations  the  following  important  properties 
can  be  established.  Let  x,./M»  be  written  zjj^  to  indicate  its  dependence 
upon  T.  Then 

0  (»-l *),  (19) 


where  m  is  any  integer;  for,  the  coefficients  x^,  etc.,  are  those  of  the 
terms  which  do  not  involve  An, ,  w, ,  and  rt ,  these  terms  being  obtained  by 
putting  An,  =  co,  =  T,  =  0  in  the  developments.  But  for  these  parameters  equal 
to  zero  the  initial  positions  are  apses  and  the  periods  (in  0  are  2r/n(. 
Hence,  at  T  =  mTr/q,,  Mt  is  at  an  apse  and  x'l  =  wt  =  Q,  whatever  the  value 
of  ft.  Since  this  is  true  for  a  range  of  values  of  et,  it  follows  that  the  coeffi- 
cient of  each  power  of  e<  in  z,'  and  in  wt  is  zero  at  T  =  mr/g,. 

It  is  evident  that  in  the  terms  independent  of  M  only  those  parameters 
appear  whose  subscript  is  the  same  as  that  of  the  coordinate  developed ;  the 
terms  involving  /i  introduce,  however,  the  other  4(A  — 1)  parameters. 

Terms  involving  p. — The  only  terms  involving  M  whose  coefficients  are 
needed  in  the  sequel  are  n  and  pe,  (j-1,  .  .  .  ,  k).  Let  the  coefficient  of 
n  in  the  development  of  x,  be  z,(0;  T),  and  that  of  ne,  be  z«(j;  T);  let  the 
coefficients  of  the  same  quantities  in  wt  be  respectively  w,(0;  T)  and  w,(j;  T) 
(t=l,  .  .  .  ,k;j  =  l,  .  .  .  ,k).  The  process  of  finding  these  depends  as  fol- 
lows upon  two  properties  of  the  solutions: 

•Moulton,  Introduction  to  CeUttial  Mechanic*  (new  edition),  p.  171. 


432  PERIODIC   ORBITS. 

(a)  Since  the  solutions  must  satisfy  the  differential  equations  identi- 
cally in  the  parameters,  the  equating  of  coefficients  of  corresponding  powers 
on  both  sides  furnishes  sets  of  differential  equations  for  the  successive  co- 
efficients in  the  solutions. 

(6)  The  arbitrary  constants  which  the  successive  coefficients  carry  are 
determined  by  the  conditions  that  the  solutions  shall  reduce  identically  to 
equations  (17)  at  r  =  0. 

For  each  pair  of  coefficients  xf(f;  r)  and  wt(f;  r)  (/=0,  .  .  .  ,  K), 
equations  (7)  give  two  simultaneous  differential  equations  of  the  second  order. 
The  one  from  (7a)  can  be  integrated  once  immediately,  and  its  integral 
combined  with  the  equation  from  (76)  renders  the  latter  a  well-known  type, 


m-O 


(20) 


Its  solution,  z((/;  T),  when  substituted  into  the  first  integral,  permits  the 
final  integration  for  wt(f;  T).     The  initial  conditions  are 

xt(f;0)=x'(f;0)=wt(f;Q)  =  w'i(f;0}=0        (i=l,  .  .  ,  fc;/=0,  .  .  .  ,  *),     (21) 

for  the  conditions  (17)  do  not  involve  /*  at  all. 

Now  the  form  of  the  solution  varies  greatly  according  as  a  term  COS$,T 
or  sin^T  is  or  is  not  present  in  (20).  In  the  former  case  the  solutions  con- 
tain a  so-called  Poisson  term,  TCOsg,T  or  r  sin  q,r,  and  in  the  latter  case  they 
do  not.  In  all  the  xt(f;r)  and  Wt(f;r)  (/  =  !,...,  /c),  a  Poisson  term  is 
present;  they  are  present  in  the  Z,(O;T)  if,  and  only  if,  for  some  pair  of  the 
HI  ,  say  n,  and  n,  ,  there  exists  an  integer  J  such  that 

/•  (H,-n.)=nf.  (22) 

The  meaning  and  consequences  of  such  a  relation  will  be  discussed  in  §219. 
In  performing  the  integrations  it  is  necessary  to  expand 

(  1  -  2ew  cos  4>fl  +  «,*  )  ~5  («  =  3,  5) 

as  a  cosine  series,  where,  for  the  sake  of  a  uniform  notation,  the  following 
definitions  are  made: 


e0  =  a,,     and     77y  =  o/,,  if  j  <i',  eu  =  a0     and     f]tl  =  1,  ifj>i.         (23) 

Then 


(24) 

00 

=   2   (?„,(€(,)  cos  m  </>;,, 

m-O 

where  the  Fm  and  Gn  are  well-known  power  series  in  eu,  beginning  with  e™- 


PROBLEM   OF  k  SATELLITES.  433 

Finally,  the  desired  coefficients  of  the  functions  x,  and  to,  are,   for 
r'=l  .....  A   and  fnr/  =  ().  1  .....  /,- 


, 


+K,,T*\n2qlT+  2  (a^' 


(25) 


—  2E,fTS\nq,T  — 


r  • 

+-K,,Tcos2q,T+  2  (frT/al 


where  />»  =  ()  and  J(/  =K,,  =  0  for/^t,  while  if  every  ^  =  0  or  T,  then 

y«=o,  c(/=A/=£,/=/>1/=c17=C=o      </-o,  ...,*);    (26) 

but  if  no  relation  (22)  holds,  then 

£„  =  //«  =y(,  =  A',,  =  0,  I),,  =  0  (/  =0  .....  Jk). 

Those  constants  whose  values  are  needed  in  the  proofs  are 

//„=  -  '' 


(27) 


i       «-i 
\\  here 


»•  i 


2S 


where  the  A",,  and  Yu  vanish,  except  for  special  relations  among  the  q,. 


434  PERIODIC    ORBITS. 

215.  Existence  of  Periodic  Orbits  of  Type  I. — When  k  finite  satellites 
are  subjected  to  the  initial  conditions  (17),  the  solutions  of  the  differential 
equations  are  expressible  as  power  series — whose  first  coefficients  have  been 
tabulated  in  equations  (18)  and  (25) — in  n  and  the  An,,  c{,  ut,  and  T,,  con- 
verging throughout  an  arbitrarily  preassigned  time  interval  for  sufficiently 
small  values  of  these  4/c+l  quantities.  And,  although  the  orbits  are  in 
general  non-periodic  (as  shown  by  the  non-periodic  terms  in  the  tabulated 
coefficients),  it  will  now  be  proved  that  the  conditions  for  periodicity  can 
nevertheless  be  satisfied,  provided  that  every  X,  =  0  or  TT,  by  assigning  to  the 
w,  and  Ti  the  value  zero  (identically  as  to  M)  and  to  the  A«(  and  c(  certain 
values  dependent  upon  /u,  that  is,  at  r  =  0  there  is  to  be  a  symmetrical  con- 
junction in  which  the  velocities  and  distances  must  be  properly  chosen 
with  reference  to  the  masses  if  periodic  motion  is  to  result. 

In  this  case  the  conditions  for  periodicity  are  (16),  of  which  equations 
(a)  are  already  satisfied.  The  necessary  and  sufficient  conditions  are  then 


(29) 


(a)       Q  =  &ni(qii 

(6)       0=  e((9,sin<?,7r)+M(x;(0;7r))+An,.p(<noo(7r) 


2 


The  problem  of  solving  these  2k  equations  has  been  treated  in  Chapter  I. 
Here  the  functional  determinant,  taken  at  An,  =  e(  =  /j.  =  0,  is  merely  the  deter- 
minant of  the  linear  terms,  whose  value  is 

t 

,  (30) 


so  that  its  vanishing  depends  upon  the  qt  .     Since,  from  (2)  and  (5), 


it  follows  that  At  vanishes  if,  and  only  if,  the  arbitrary  qt  is  selected  as  an 
integer. 

Case  I:  qt  is  not  an  integer.  —  Since  Aj^O,  the  An,  and  et  exist*  as  unique 
power  series  in  /i,  vanishing  with  n,  converging  for  /j,  sufficiently  small,  and 
satisfying  (29).  Thus,  for  all  sufficiently  small  masses  of  the  k  finite  satel- 
lites there  exist  initial  conditions  for  which  the  resulting  orbits  are  periodic; 
and  the  presence  of  qt  and  of  the  arbitrary  integers  pt  shows  that  for  given 
masses  there  is  a  Axfold  infinitude  of  orbits  of  this  type.  A  family  of  such 
orbits  exists  "growing  out  of"  any  set  of  circles  for  which  the  infinitesimal 
system  would  have  commensurable  synodic  periods,  whose  least  common 
multiple  is  not  divisible  by  the  sidereal  period  of  satellite  k.  In  Case  I  consecu- 
tive symmetrical  conjunctions  do  not  occur  at  the  same  absolute  longitude; 
nor  will  any  later  ones  occur  at  the  same  longitude  if  qt  is  irrational.  f 

•See  §§1,  2.  fSee  §219. 


I'HDItl.KM    (IF    A'    SA'I  KI.l.l  I  I  .-.  !.V> 

H:  ,it  i*,tn  integer.—  Here  A,  =  ().      Nevertheless  the  .lacohian  of  the 
with  respect  to  the  A/>;,  taken  at  An,  =  e,  =  p  =  0,  is 

A^T'M^O.  (31) 

(The  only  possibility  for  qt  =  0  is  p,  =  -qt  which  requires  n,  =  0,  and  such  a 
-election  for  p,  has  been  excluded.)  Hence  (29a)  can  be  solved  for  the  A;/,* 
as  power  series  in  the  e,  and  n,  converging  for  sufficiently  small  values  of  the 
latter  (plant  it  ies.  Now,  by  (19),  every  term  in  (29a,  6)  has  either  n  or  some 
±11,  as  a  factor:  hence  the  solutions  have  the  form 

A//,=M/'((C;,M)  (»'-!  .....  t;  j-l  .....  k).        (32) 

If  the  power  >eries  (32)  are  substituted  in  (296),  the  resulting  series  converge 
for  sufficiently  small  values  of  M  and  e,  and  contain  n  as  a  factor.  (This 
merely  means  that  AM,=M  =  O  satisfy  the  periodicity  conditions,  whatever 
may  be  the  values  of  the  e,).  If  p  is  divided  out.f  relations  are  obtained 
among  the  e,  and  n  of  the  form 

' 


At  this  point  two  questions  of  importance  arise,  viz.,  as  to  the  vanishing 
of  the  x',(0;  T)  and  as  to  the  vanishing  of  the  determinant  of  the  coefficients 
of  the  linear  terms  in  e,.  By  (25)  and  (27)  every  x',(Q;  r)  is  zero  unless  the 
relation  (22)  holds  for  some  pair  of  the  n,.  When  such  a  relation  does 
hold,  the  equations  (33)  are  not  satisfied  by  f,  =  /u  =  0,  so  that  solutions  for 
the  c,,  vanishing  with  p,  do  not  exist.  Hence,  periodic  orbits  of  Type  I, 
"growing  out  of  the  circular  orbits,"  do  not  exist,  if,  for  any  n,  and  n,, 
J  •  (n/—  n,)  =  n,  ,  where  J  is  an  integer.  When  no  such  relation  exists,  J 
the  equations  (33)  are  satisfied  by  e,  =  n  =  Q.  It  remains  to  examine  the 
determinant  A,  of  the  first  degree  terms  in  the  e,.  This  involves,  in  each 
of  its  elements,  power  series  in  the  to,  or  (it/a,,  a,/o«;  and  it  is  unknown 
whether  there  are  any  sets  of  values  of  the  e«/  for  which  A,  =  0.  It  will,  how- 
ever, be  shown  that  there  is  an  infinite  number  of  values  for  which  the 
determinant  is  distinct  from  zero. 

Let  P«/  be  any  element  of  A,,  the  first  subscript  indicating  the  row  and 
the  second  the  column;  then,  by  equations  (33)  and  (25), 

Pl,  =  x'l(f;T)=ql(-ir>*H,f  (/*<),          1 

Pll=x'l(i;*)-ql(-l)'<wl(0;ir)=qt(-ir<*(Hll-.\l9}. 

The  «„  depend  upon  v,  qt,  the  ft,  and  the  p,  which  were  arbitrarily  chosen. 
It  will  now  be  shown  that  for  a  fixed  selection  of  the  0,,  v,  and  qt  there  is  an 


This  cane  U  not  essentially  different  from  that  treated  in  |4.  For  although  Ai  haa  no  minor*  of  order 
leas  than  fc  distinct  from  icro,  yet,  so  far  as  the  linear  terms  alone  are  concerned,  the  equations  (29)  may  be 
regarded  as  k  independent  pairs. 

fit  is  precisely  this  step  which  makes  the  selection  of  the  parameters  4ft, ,  r,  especially  advantageous. 

{That  qt  an  integer  does  not  involve  the  existence  of  such  a  relation  is  shown  in  §2 in. 


436  PERIODIC    OUBITS. 

infinite  number  of  selections  of  the  pf  (viz.,  all  for  which  the  e,;  are  "suffi- 
ciently small")  for  which  A4J  =^0.  For  convenience,  let  all  the  ct,  be  expressed 
in  terms  of  a  single  parameter  a  by 

at  =  6X(t-°at  «=  l,  ____  *-  1),         (35) 

where  the  b,  ,  v,  and  g*  (hence  also  ak  and  ??»)  are  constants  independent  of  a. 
Every  element  of  A3  is,  then,  a  power  series  in  a;  for  by  (35),  (34),  and  (27) 


62 
; 


JM-ft 


<< 


if 


"    •   Qw(a),  if/>l, 

where  the  Qv(a)  (/=  1>  •  •  •  >  &)  are  power  series  in  a,  beginning  with  a 
constant  term*  In  P«  under  the  sign  S',  the  lowest  power  of  a  for  j  <  i  is 
lOt  —  4j  —  6k,  this  exponent  having  its  smallest  value  6i  +  4  —  6/c,  when  j  =  i—l; 
while  for  j>i  the  lowest  power  is  6j  —  6/k,  whose  smallest  value  is  6t'+6  —  Gfc. 
Hence  in  the  i*  row  of  A,  the  lowest  power  of  a  in  any  of  the  Pv  is 

for/<t,  6t+6  —  6fc,  viz.,  for/=i—  1, 

for/=l,  6t+4  —  6fc,  except  when  i  =  1, 

for/>i,  6i+8-6/c,  viz.,  for/=i+l. 

But,  in  the  first  row  Pu  carries  a11""  as  against  a14"'*  in  Pu,  which  is  the  next 
lowest  power.  Evidently,  then,  in  every  row  of  A3  the  lowest  power  of  a 
occurs  in  the  element  of  the  main  diagonal;  and  if  that  lowest  power  of  a 
is  removed  from  the  row  as  a  factor  of  A3  ,  a  new  determinant  A4  is  obtained, 
all  of  whose  main  diagonal  elements  begin  with  a  constant  term,  while  the 
series  in  every  other  element  carries  a  positive  power  of  a  as  a  factor.  Thus 
the  development  of  A4  as  a  single  series  in  a  begins  with  a  constant  term  (the 
product  of  those  in  the  principal  diagonal)  and  is  distinct  from  zero  both 
for  a  =  0  and  for  all  values  of  a  up  to  some  finite  value.  Hence  A3  also  must 
be  distinct  from  zero  for  all  values  of  a  sufficiently  small,  and  vanishes,  if 
at  all,  at  a  finite  number  of  points. 

In  case  some  <?/  =  3qt  ,  the  special  terms  A',/  and  YtJ  (which  may  be 
present  in  the  Pv  and  P«  for  other  special  relations  also)  carry  powers  of  a 
lower  than  some  of  those  considered  above.  In  Pv  the  power  is  lowered  to 
aIOi~4/~",  and  in  P«  to  asl+4~";  in  P/(  to  aM~",  and  in  Pn  remains  unchanged  as 
a«/+4-et  guj.  jn  everv  case  ^  is  easily  seen  that  the  power  is  lower  in  the 

*If  X(f  and  YU  are  present  in  P,/  and  PH,  certain  changes  must  be  made. 


PHOHJ.KM    Ml     /.     -A  I  KIM  I  l>  -i;<7 

main  diagonal  element  than  elsewhere  in  the  same  row,  except  in  the  first 
n>\\,  where  /'„  may  now  carry  aa~u,  as  does  Pu.  And  even  this  exception 
is  immaterial,  since,  when  the  rows  have  been  factored  as  before,  the 
constant  term  in  the  element  Qu  is  to  be  multiplied  by  a  minor  whose  first 
column  contains  a*  as  a  factor,  and  hence  can  not  destroy  the  constant 
term  in  the  main  diagonal  product.  Therefore,  it  is  true  without  exception 
that  Aj?* 0  for  all  values  of  a  sufficiently  small. 

Therefore,  equations  (33)  can  be  solved  for  the  e,  in  terms  of  n,  vanishing 
with  n.  The  substitution  of  these  solutions  in  (32)  gives  the  An<,  as  well 
as  t  he  e, ,  as  holomorphic  functions  of  /*,  for  ju  sufficiently  small.  Hence  there 
exist  initial  conditions  giving  periodic  orbits  of  Type  I  even  when  qt  is  an 
integer,  provided  that  no  relation  (22)  holds  and  that  a  is  sufficiently  small. 

In  drawing  this  last  conclusion,  however,  a  point  of  delicacy  arises. 
The  p,  are  functions  of  a  in  the  foregoing  argument,  and  obviously  the  p, 
are  not  integers  (as  the  formulation  of  the  problem  requires  them  to  be) 
for  all  values  of  a  on  any  interval.  The  question  arises  as  to  whether  there 
are,  indeed,  any  values  of  a,  "sufficiently  small,"  for  which  the  p,  are  integers. 
From  (36)  and  pt  =  qt—qt,  it  follows  that 

l)  0-1 t-1).      (37) 

The  present  discussion  will  be  confined  to  exhibiting  a  selection  of  the  l>, 
such  that  there  is  an  infinite  number  of  values  of  a  less  than  any  assigned 
quantity,  for  each  of  which  the  pt  are  integers  without  common  divisor. 
Let  the  assigned  value  be  a,  and  let  the  6»  be  defined  by 


(i"  .....  *-|)!  (38) 

and  consider  (37)  for  a  =  a9-\/l/n,  where  n  is  an  integer.      Evidently,  since 
t  is  an  integer,  and  since 

-qt         0-1  .....  fc-1),      (39) 


'- 


the  p,  are  integers.  Consider  the  possibility  of  a  common  factor.  If  p,_, 
and  p»_,  have  a  common  factor,  their  difference  has  the  same  factor.  Thus, 
if  there  is  a  factor  common  to  (g*+l)n-g»  and  (g*+2)n*-g»,  it  is  also  a 
factor  of  n*+n(n  —  !)(&+  1).  Hence  if  n  is  prime  and  greater  than  qt, 
such  a  factor  must  divide  n-|-(n—  l)(g»+l)  and  also  the  difference  between 
this  number  and  p»_,,  or  (n—  1).  But,  as  n  and  n—  1  are  mutually  prime, 
there  is  no  factor  of  n  —  1  which  divides  n+(n  —  1)  (</»+!),  and  hence  no 
factor  common  to  p»_,  and  p»_,  if  n  is  chosen  a  prime  number  greater  than  qt  . 
There  is  an  infinite  number  of  primes;  hence  the  p,  have  the  stated  property. 
The  periodic  solutions  exist,  then,  and  might  be  obtained  as  scries 
in  n  alone  (convergent  for  sufficiently  small  values)  by  substituting  in  the 
original  series  in  »  and  the  An,  and  et  the  values  of  the  latter  2k  parameters, 
as  obtained  in  terms  of  M  from  the  periodicity  conditions.  A  far  more  advan- 
tageous method  is,  however,  available. 


438  PERIODIC    ORBITS. 

216.  Method  of  Construction  of  Solutions.   Type  I.  —  It  has  been  shown 
that,  for  M  sufficiently  small,  there  exist  series 


(T)  n"  (i  =  l,...,  fc),  (40) 

n=l  n  =  l 

which  (a)  converge  for  0  ^  T  ^  Zir,  (b)  satisfy  the  differential  equations 
(7),  (c)  satisfy  x\  (0)  =  0,  wt  (0)  =  0  identically  in  n,  and  (rf)  satisfy 
a-,(r  +  27r)  —X,(T)  =  wi(T+2ir)  —  W,(T)  =0  identically  in  /*. 

The  permanent  convergence  of  series  (40)  follows  from  (d)  and   («). 
From  (c)  and  (rf)  follow  respectively 

<»(0)=^,B(0)=0,  (41) 

x,,n(T  -\-2rr)  -xt,n(r')=wi,,(T  +  2ir)  -W,..(T)  =0.  (42) 

These  equations  (41)  and  (42)  will  determine  the  constants  of  integration 
arising  at  each  step. 

First  order  terms.  —  Since   (40)   must  satisfy   (7)  identically   in  n,  the 

.TM(T)  and  W>(,I(T)  must  satisfy 

(a)        <1+2g<<1+S'«wsin^l  -  tf,  I  F.(O*»«fe        =  0, 


(6) 


2 
y        L  m 


F^e^cosnxb  J  =0. 
=o 


(43) 


Since  every  <j>it  is  a  multiple  of  T  plus  a  multiple  of  IT,  equations  (43)  are  of  the 
typo 

(a)  «>",  +  2?X.+  SDftsramr-0, 

(44) 


where  the  D'™'  arid  ^iTi  are  linearly  related  to  the  Fm,  and  can  be  ex|>rossed 
in  terms  of  the  latter  as  soon  as  the  pt  are  chosen.     The  solutions  are 


S  A*  cos  WIT, 


where  the  c|*  (j  =  1,  .  .  .  ,  4;  t  =  1,  .  .  .  ,  fc)  are  the  constants  of  integration  and 

(»»'  -^D-4  M  =  #M  -  ^  DM,         ^2^'"'  =  ^M  -  2wi9,  ^1  M.  (46) 

A  *6 

Poisson  terms  do  not  appear  in  (45);  for,  since  no  relation  (22)  holds,  no 
term  in  cosq,r  or  sin^T  is  present  in  (44).     Now,  by  (41)  and  (42), 

~<3>  _  „<<>  _  A  /-("  -       —  F™ 

°l,l        t/(,l~u»  CM  OQ   -"lill 

so  that  the  A;  constants  c<"  alone  remain  to  be  determined.      And  here 
arise  two  cases,  just  as  in  the  existence  proof: 


PROBLEM    OF   k   SATELLITES. 

Case  I.  qt  is  not,  an  integer.  —  Here,  q,  not  being  an  integer,  coeq,  r  does 
not  have  the  period  2*;  consequently,  by  (42),  c"  =  0  (t  —  1,  .  .  .  ,  k). 

('use  II.  qt  is  an  integer.—  Here  cosqtr  has  the  period  2ir,  and  (42)  is 
satisfied  for  the  arbitrary  c.^O.  These  k  Constanta  remain  undetermined 
until  the  second-order  terms  are  found,  when  the  c*  are  uniquely  determined 
in  destroying  Poisson  terms. 

Terms  of  any  order.  Case  I.  —  Assume  that  for  n-1,  .  .  .  ,  h—\, 
the  x,,n  (T)  and  w,..  (T)  have  been  found,  the  constants  being  determined,  and 
have  the  form 


xlin(r)=  2  ^«coa»wr.         W,..(T)-  2  B^rinmr         (»-!,...,*).       (47) 


An  induction  will  show  that  X^(T)  and  u>(,»(r)  have  the  form  (47);  that, 
moreover,  the  differential  equations  for  x,,»  and  u\*  are  of  the  type  (44);  and 
that  the  constants  of  integration  are  determined  just  as  in  the  preceding  case 
for  n  =  1.  The  differential  equations  are  [see  (7)] 

(a)  2  .T,tu>",+2  2  .r;..tc;.,+2g,jr;.. 


(b)         a-;;-     2     *l* 

t+l+m-k 


+qtltil(x7\-l+2'8l,>alj  2 


(48) 


+       2      Xj^cos^+Wj-w^xT'-ffu1}^      =0, 

*+«+•-»-! 

where  an  expression  in  parenthesis,  having  a  subscript  /  outside,  denotes  the 
sum  of  all  those  terms  in  the  expression  which  involve  /*'•  It  is  to  be  shown 
(a)  that  the  variables  in  (48)  whose  second  subscript  is  h  enter  in  the  same 
form  as  the  x<,i  and  w,,i  enter  (44),  and  03)  that  the  remaining  terms  of 
(48)  (a)  and  (6)  reduce  respectively  to  a  sine  series  and  a  cosine  series  in 
multiples  of  T. 

Evidently  in  (48a)  the  only  terms  involving  the  x,.»  and  i0,,»(t'=  1,  .  .  .  ,  h) 
are  w"Jt+2qtx'IJt;  and  in  (486),  aside  from  (x,'1).,  they  are  x't'JI-2qlw'IJt-qttx^. 
Now  it  can  be  shown  easily  by  induction  that 


«r—  (2;)'', 

*•  V«M    'M-0 


where  the  N,  are  positive  numbers  and  the  v,  are  jwsitive  integers  (or  zero) 
satisfying  the  conditions 


Now  xljt  enters  only  through  (efz./d//)'*,  and  for  this  term  Nr=2,  vt=  1, 
vk=  1,  and  »f  =  0(f=  I,  •  .  .  ,h-l);  hence,  in  (xr')»,  xljt  appears  with  the 
coefficient  -  2.  Therefore,  in  (486),  the  terms  involving  xljt  and  wtjt  are 
x't'jt-2qtwfljt-3qtlxtj>,  and  this  establishes  statement  (a)  above. 


440  PERIODIC   ORBITS. 

On  using  the  notation  F'(T)  and  F'(T)  to  designate  respectively  a  cosine 
series  and  a  sine  series  in  multiples  of  T,  it  is  evident  that 


(F')n  =  Fe,        (FT  =  Fe,      (FT+l  =  F'  . 

Hence,  as  every  x,,n  and  w'tM(n=l,  .  .  .  ,  h-1)  is  a  Fe(r},  so  also  are  all 
sums  of  products  of  these  quantities,  and  also  all  polynomials  in  the  xiM  ,  e.  g. 
Orx),  and  parts  of  (o--3),.  Similarly,  the  sums  of  all  products  xttiw"t  and 
or'.X.i  are  F'(T). 

There  remain  in  (48)  only  the  terms  (sin  mv+Wj  —  Wt).  where  w(>  =  <£;,  in 
(48a)  and  m(J  =  <^,  +  7r/2  in  (486).  Let,  for  the  moment,  ztj  =  mtJ+Wj  —  wt. 
Then,  since  [dfzij/dnr]lt.0  =  wj.f—  wtif,  it  can  be  shown  by  a  simple  induction 
that 


where  the  N,  are  numbers  and  the  vf  satisfy  (49)  after  h  is  replaced  by  /. 
Now  since  every  (w}.,  —  wt,f)  (/  =  !,  .  .  .  ,  h—\)  is  a  F'(T),  the  product 
^'[(Wj.f  —  w^  is  a  F'(r)  or  F°(T)  according  as  ^'t~\v,  is  odd  or  even.  But 
when  this  sum  is  odd,  vt  is  odd;  and  when  even,  i>0  is  even.  The  entire 
product  n£o  is  then  always  a  F'(T]  if  mt)  =  <j>]t,  or  a  FC(T)  if  mu  =  01(+7r/2. 
The  differential  equations  (48)  are,  therefore,  of  the  form 


(a)  M&  +  2qtx'(A  +  S  D«  sin  mr  =  0, 

m  =  l 

(b)  a-,',;-2<7X,»-3^,,+^+  S  ^ 


m-l 


(50) 


where  no  term  in  cos  qtT  or  sin  g,r  occurs  under  the  summation  sign,  since  q, 
is  not  an  integer.  Obviously  the  integration  of  (50)  is  the  same  problem  as 
that  of  (44),  so  that  the  solutions  are 

2  A 


+  2 

m-l 

«  =  !, 

where 

Crw1  —  n2^  4  <ro)  —  ft1""'  —  2<7<  n<"»  ii)2R(m)  —  D0"' 

<?iM(,»  —  -°i.»      m     •*  '        mLii,i>—Lft,h 
Also,  by  (41)  and  (42), 

r(1)—  -  —  ff"o)         />(2)  —  /.«'  —  /.(4)  —  n 
(~  •Cl  r    —  c    —  c    - 


PROBLEM    OF   k    SATELLITES.  441 

so  that  the  induction  is  completely  established.  Thus  the  successive  z(- 
and  wtM  reduce  to 

*  +24 £  cosmr,     WIM(T)  =  2    B% sin  mr    « -  1 fc) ,  (5 1') 

and  may  be  obtained  without  inhynitiim  by  applying  (52).  It  is  merely 
necessary  to  compute  the  D™  and  E£  from  equations  (48)  at  each  step. 

T>  rms  of  any  order.  Case  II. — Assume,  as  in  Case  I,  that  for  n=  1,  .  .  .  , 
(/»-!)  the  Z,..(T)  and  WIM(T)  have  the  form  (47),  all  the  constants  of 
integration  having  been  determined  except  the  cJJ|_,.  The  differential 
equations  for  the  xljk  and  wljt  are  again  (50)  (a)  and  (6),  where,  however, 
since  (jt  is  an  integer,  cos  qt  T  and  sin  q,  T  may  occur  under  the  summation  sign. 
These  terms  arise  from  twosources:  from  the  terms  c*_,  cos  q,  rand  c^_,  sing,T 
in  the  z,.»_,  and  u>,.»_i,  and  from  similar  terms  in  the  earlier  x,M  and  u'l-t  as 
well  as  (usually)  from  combinations  of  the  <t>,,  in  the  coefficients. 

When  (50a)  are  integrated  and  combined  with  (506),  equations  for  the 
z(JI  are  obtained;  to  avoid  Poisson  terms  in  the  xljt,  the  coefficients  of  the 
terms  in  cos  q,  T  in  these  last  equations  must  be  made  to  vanish.  These  co- 
efficients, E%  —  2D'tf,  involve  the  c^_,  and  various  known  constants;  e.g., 
the  c™  (n  =  1,  .  .  .  ,  h-2);  and  the  vanishing  of  the  E%  - 2 D%  will  usually 
determine  the  c^_, . 

In  the  first  place,  the  only  terms  of  (48)  in  which  the  j-, ,_,  and  H'(Jk_, 
appear  are* 

(a)    *,,_,» 


.  I  , .    ^  x 

+  26,,  •{  —  (2zM_,cos</>/(+ifM_,  —  Wut-iSmt,,) 

i        I 

,  sin  <!>„) 


>>()  (aj/r,.,., +ZM_,  -  airrM_, +*,.._,  cos  </>;, 


(54) 


•For  A  -2  the  first  four  U-niu  in  (a)  and  in  (b)  »re  to  b«  divided  by  2. 


442  PERIODIC    ORBITS. 

where  <riji0  =  (1  —  2e(,  cos<^(  +e^)l/*.  Whence  it  is  seen  that  the  c™^  enter  the 
E^  —  2D^  linearly;  and,  denoting  their  coefficients  by  Rif(i=l,  .  .  .  ,  k; 
/=  1,  .  .  .  ,  k),  it  is  found  that 


where  the  Pif  are  the  elements  of  the  determinant  A3,  discussed  in  the 
existence  proof.  Hence,  when  A3?^0,  the  determinant  of  the  coefficients  of 
the  c®_!  in  the  equations  E%  -  2D^  is  zero.  The  constants  c£»_,  (/=!,...,  /c) 
are  then  uniquely  determined. 

The  values  of  the  xt,h(r)  and  with(r)  are  given  by  (51)  and  (52)  ;  equations 
(53)  are,  however,  replaced  by 

CM=-     -  -  Ew)  r(3)-r(4)—  f)  (V-1  frl  (VVl 

'*  3(7      '•*'  c(,»~c(,»~  (.1—  J,  .  .  .  ,  «;, 

the  c,("  remaining  undetermined  at  this  step.  The  induction  is  thus  estab- 
lished. In  this  case,  too,  the  successive  xttn  and  wttn  have  the  form  (51'),  the 
c'"  appearing  as  A%£,  and  are  obtained  without  integration.  But,  besides 
computing  the  D™  and  E%  and  applying  (52),  it  is  necessary  to  obtain 
also  the  D,(?n+i  and  ^?J+i  and  to  determine  the  cj"  by  solving  the  equations 

r/>(«,>         9D"'5     —  fl 

'il.n+l  —  *M.n+l- 

Remarks:  (1)  In  constructing  the  solutions  it  has  been  tacitly  assumed 
that  the  Fourier  series  representation  of  the  x<M  and  wfitt  and  the  attendant 
manipulations  of  these  series  are  valid.  This  is  justified  by  the  considera- 
tion that  the  well-known  functions  encountered  in  the  first  step  are  repre- 
sentable,  together  with  their  derivatives,  by  uniformly  convergent  Fourier 
series.  But  the  products  of  two  such  series  is  another  of  the  same  type,  and 
also  the  integral  of  a  uniformly  convergent  Fourier  series  is  uniformly  con- 
vergent. At  every  step  the  convergence  remains  uniform. 

(2)  The  question  naturally  arises  as  to  whether,  in  any  orbits  of  Type  I, 
the  smallest  period  is  a  multiple  of  the  period  of  the  infinitesimal  system, 
namely,  mT.  An  examination  of  the  method  of  constructing  the  solutions 
furnishes  the  answer.  If  mqt  is  not  an  integer,  the  constants  are  determined 
precisely  as  in  Case  I;  if  mqt  is  an  integer,  then  by  the  method  used  in  Case  II. 
In  any  event,  for  a  given  value  of  n  and  a  given  set  of  pt  and  qk,  there  is 
a  unique  solution  satisfying  (41)  and  (42)  (with  2rmr  replacing  2ir).  The 
solutions  found  above,  however,  satisfy  these  conditions;  hence  there  are  no 
orbits  of  this  type  whose  smallest  period  is  a  multiple  of  the  period  of  the 
infinitesimal  system. 

217.  Concerning  Orbits  of  Type  II.—  For  Type  II,  as  for  Type  I,  all  the 
X,  are  multiples  of  TT;  but  it  is  proposed  to  ascertain  whether  the  co,  and  r, 
exist  as  functions  of  /x  (not  identically  zero,  but  vanishing  with  /u)  so  as  to 
satisfy  the  periodicity  conditions  (13).  The  question  can  be  studied  most 
easily  by  examining  the  method  of  constructing  the  solutions. 


PROBLEM    OK    A'    SATKU.ITB8.  I  \'.\ 

Let  the  origin  of  time  be  selected  as  the  instant  when  satellite  k  has  an 
apsidal  passage,  and  the  origin  of  longitude  at  that  satellite's  apsidal  position 
so  that  «t  =  T»  =  0;  and  let  it  be  assumed  for  the  moment  that  the  w,  and  T, 
(i=l,  .  .  .  ,  k  —  1)  exist  as  functions  of  »  satisfying  (13);  then  there  e\i-r 
solutions  of  the  type  i  10)  satisfying  (42),  but  not  satisfying  (41),  for  all  values 
of  n  except  for  i  =  k.  Because  of  the  absence  of  conditions  (41)  some  of  tin* 
constants  in  each  step  are  left  undetermined  until  terms  of  higher  orders  are 
found. 

For  the  first-order  terms  the  differential  equations  are  again  (44),  and 
the  solutions  are  (45),  the  coefficients  being  determined  by  (46).  Evidently 
the  c",}  are  determined  as  in  Type  I,  but  for  the  other  c™  two  cases  arise. 

Case  I.  qt  is  not  an  integer. — Here,  by  (42),  cj^  =  c"  =  0  (t  =  1,  .  .  .  ,  A;), 
but,  the  cf^  are  at  present  undetermined,  except  c£|  =  0. 

Case  II.  qt  is  an  integer. — Here  the  terms  cosg(r  and  &\nqtr  have  the 
required  period  2r,  and  hence  the  c",'  and  c™,  together  with  the  c™,  remain 
undetermined,  except  c£{  =  c£J  =  0. 

Terms  of  any  order.     Case  I. — Consider  next  the  terms  of  order  two.    The 
differential  equations  are  still  (48),  using  /t  =  2;  however,  because  of  the  c",' 
equations  now  reduce  not  to  (50),  but  to  the  form 


(«) 

(57) 

(W 


where  the  //£*  and  ./£'  vanish  with  the  c£|.  The  first  integration  apparently 
introduces  non-periodic  terms  tif^r;  but  computation  shows  that  these  coeffi- 
cients vanish.  Thus 


and  by  Le  Verrier's  relations  among  the  Fn(etl),  and  G'.fo,),*  it  is  found  that 
the  bracketed  expression  vanishes  identically  in  the  tir  Hence  the  solutions 
of  (57)  have  the  form 


(58) 


where  the  P*'  and  Q£  vanish  with  the  c%,  which  (together  with  the  c£) 
remain  at  present  undetermined,  the  other  constants  arising  in  the  x(4  and 

•Ti-werand:     Mfeanujuf  CHrrit,  TO|.  I,  pp.  27'i 


r-I4;       +  2  (fljywnnjT+gjJoosf/iT). 


444  PERIODIC   ORBITS. 

the  wlA  having  been  determined  as  were  the  corresponding  ones  in  the  first- 
order  terms. 

For  the  terms  of  order  three,  the  differential  equations  are  of  the  form 
(57),  and  the  H™  do  not  vanish  identically.  For,  while  the  c$  drop  out  just 
as  the  c"J  did  in  the  preceding  step,  the  c™  are  now  present  (linearly  only), 
entering  both  through  the  P™  and  Q™  and  directly  from  the  W}<I(T)  .  The 
destruction  of  the  non-periodic  terms  H™  T,  by  setting  the  H™  =  0,  will  be 
found  to  require  the  vanishing  of  these  c(%  which  have  been  previously  unde- 
termined. 

The  Hfj  consist  of  two  sorts  of  terms:  those  which  arise  identically  by 
combining  trigonometric  functions  of  a  single  <j>it  and  its  multiples,  and  those 
which  may  appear  because  of  relations  among  the  various  $„.  Terms  of 
the  latter  sort  can  not  be  collected,  unless  the  numerical  values  of  the  pt 
have  been  chosen;  but  their  possible  presence  will  be  shown  to  be  immaterial 
so  far  as  the  conclusions  are  concerned. 

In  equations  (48a),  using  h  =  3,  the  terms  which  reduce  identically  to 
constants  may  be  selected  by  fixing  upon  some  one  <t>jt  and  expressing  the 
coefficients  of  cos  ra</>yi  in  xtA  and  wiA,  and  of  sin  m<pit  in  wtA  and  xiA  .  Incident- 
ally it  may  be  noted  that  the  series  in  each  xtM  and  wiM,  as  given  by  (47)  or 
(58),  is  in  reality  a  rearrangement  of  several  Fourier  series  in  the  various 
4>ti  .  The  resulting  complicated  constant,  involving  several  series  in  the  Fn(ctJ) 
and  £?„(«„),  can  be  treated  advantageously  by  expressing  all  the  a,,  et},  qt, 
and  8(J  in  terms  of  a  single  parameter  a,  just  as  in  (35)  and  (36).  In  the 
H®  the  coefficient  of  each  c£J,  say  Mti>  becomes  then  a  power  series  in  a; 

and  it  is  found  that  Mit  =  —  S'  M(l,  and 

i 

Mt]  =  a«+»-«.Ntj      0->t-),  Aftf  =  a"-«  .Nti       (j<i),       (59) 

where  the  Ntj  are  infinite  series  in  their  respective  etj,  beginning  with  con- 
stant terms. 

Now  the  vanishing  of  the  H®  requires  that  the  c"J  satisfy  k  linear 
equations,  which  must  be  homogeneous,  since  the  vanishing  of  all  of  the  un- 
knowns would  reduce  the  H™  to  their  values  in  Type  I,  viz.,  zero.  That  is, 


(60) 


But,  from  the  value  of  Mit  above,  it  is  evident  that  the  determinant  of  the 
Mit  is  identically  zero,  so  that  any  equation  is  a  consequence  of  the  others 
and  may  be  suppressed.  Let  the  first  equation  be  the  one  which  is  dropped. 
Then  the  remaining  k  —  \  equations  (i  =  2,  .  .  .  ,  k)  will  determine  the 
cJa  (j  =  lj  •-•••!  k  —  1)  uniquely  in  terms  of  c£{  if  their  determinant  \,  of 
order  k  —  l'm  the  Mtj,  is  distinct  from  zero.  From  (59)  the  lowest  powers  of 


PKOBLEM    OK    /.'    v\IKI.I.m>.  445 

a  in  the  various  elements  of  any  row  of  A,  may  be  ascertained.  Evidently 
the  exponent  4i+2j  —  6k  takes  its  smallest  value  4i+2-6Jfc  when;  =  l,  and 
its  largest  value  6i—2  —  6k  when  j  =  i-i-  while  Qj—Qk  takes  its  smallest 
value  61 +6  —  6A-  when  j  =  i+l.  A>  t his  last  is  greater  than  the  highest  value 
of  the  former  exponent,  it  is  clear  that  the  lowest  power  of  a  in  any  of  the 
M,,(j=},  .  .  .  ,t-l,7  +  l,  .  .  .  .  k  - 1)  is  4i+2  —  6k,  which  occurs  in  M(l 
and  also  in  Mtt.  Thus,  in  the  H*  row  of  \ (where  i  =  r+l),  the  lowest 
power  is  4r+6  — 6A;  and  this  appears  for  r=l,  .  .  .  ,  k  —  2  in  two  columns, 
the  first  and  (r +!)<*.  But,  in  the  last  row,  where  r  =  k—  1,  the  lowest 
power  can  appear  only  in  the  first  column.  Hence  if  the  factor  a*'+4~**  is 
removed  from  the  elements  of  the  r1*  row  (r=  1 k  —  l),&  new  deter- 
minant \  is  obtained  in  whose  first  column  the  series  of  each  element 
beiiins  with  a  constant  term,  as  does  also  the  series  of  one  other  element  in 
each  row  except  the  last  row. 

Now,  if  A,  is  developed  by  the  minors  of  its  last  row,  it  is  clear  that  a 
constant  term  can  not  be  lacking  when  the  development  is  rearranged  as  a 
single  power  series  in  o.  For  the  only  element  of  the  last  row  which  can  con- 
tribute to  the  constant  term  is  that  in  the  first  column;  and  in  the  minor  of 
this  element  constant  terms  are  present  in  all  the  elements  of  the  main 
diagonal,  and  nowhere  else. 

Thus  A, ^0  at  a  =  0;  and  hence  A,  and  likewise  A,  are  distinct  from  zero 
for  all  values  of  a  sufficiently  small.  Therefore  the  vanishing  of  the  H™ 
determines  the  c£|  0  =  1  >  •  •  •  »  k— 1)  uniquely  and  homogeneously  in  terms 
of  r£J .  But  this  latter  constant  is  to  IMJ  put  equal  to  zero  by  reason  of  the 
choice  of  the  origin  of  time,  as  noted  above  in  discussing  the  first-order  terms. 
Hence  every  c%  =  0.  Thus  the  WIA(T)  reduce  to  the  values  which  they  would 
have  for  orbits  of  Type  I.  Then  also  every  P*'  =  Q*'  =  0 ;  and  the  xtA  (r)  and 
H>,.»(T)  reduce  to  the  values  in  Type  I,  except  for  the  c^,  which  remain  as  yet 
undetermined  for  j=l,  -  •  .  ,  k—1. 

In  the  next  step  these  c%  will  enter  the  H™  (by  identity)  in  precisely 
the  same  way  as  the  c%  entered  the  //",  and  must  likewise  vanish.  Simi- 
larly, the  c£  in  the  terms  of  order  n  will  remain  undetermined  until  the  //£+, 
are  set  equal  to  zero,  when  they  must  vanish  together  with  the  P™+1  and 
Q"Vu  to  which  they  will  in  the  meantime  have  given  rise.  Thus  the  final 
determination  of  the  constants  arising  at  any  step  reduces  the  terms  of  that 
order  to  the  values  which  they  would  have  for  Type  I. 

If  various  relations  among  the  p,  give  rise  to  other  terms  in  the  A/J5J 
than  those  which  are  present  identically  in  some  one  <t>)t ,  — even  if  the-' 
terms  introduce  into  various  elements  lower  powers  of  a  than  have  been 
treated  in  A»,  --  these  new  terms  can  not  affect  the  argument  in  general; 
for,  if  they  introduce  into  the  development  lower  powers  of  a  than  were 
previously  present,  with  non-vanishing  coefficients,  then  this  new  development 
is  equally  as  useful  as  the  former  value  of  A,;  while,  if  they  do  not  furni.-li 


446  PERIODIC    ORBITS. 

terms  of  lower  order,  neither  can  they  in  general  destroy  the  terms  treated 
in  A6  ,  since  they  must  involve  new  /3's  in  their  coefficients.  Any  cancellation 
could  occur,  then,  only  for  a  few  special  relations  among  the  masses. 

The  conclusion  is  not  yet  warranted  that  no  orbits  of  Type  II  exist  in 
Case  I;  but  there  can  be  none  when  a  is  below  some  ''sufficiently  small" 
finite  value,  unless  possibly  for  a  few  very  special  relations  among  the  masses. 

Terms  of  any  order.  Case  II.  —  The  differential  equations  for  the  second- 
order  terms  XIA(T),  w,,2(r)  are  again  (48);  but  because  of  the  cf\  and  c"J 
the  equations  reduce  to  (57),  as  in  Case  I  above,  where,  however,  the  H™  and 
J™  now  involve  them  (linearly)  and  vanish  with  these  two  sets  of  constants. 

In  order  that  z,,2  and  wiit  shall  be  periodic,  it  is  necessary,  just  as  for 
Type  I,  that  E^  —  2D(^,  the  coefficient  of  cos^r  in  the  final  differential  equa- 
tion for  xt,t(T),  shall  vanish.  It  is  equally  necessa^  that  J$+2Hl'f,  the 
coefficient  of  sin  q,r  in  the  same  equation,  shall  vanish.  Further,  the  "secular 
terms"  Hf^r  must  vanish.  The  H^  are  free  from  the  c"J,  just  as  in  Case  I 
above;  but  they  now  contain  the  cj\  and  c^J,  vanishing  with  the  latter  set. 

Now,  from  (54)  and  the  fact  that  no  relation  (22)  holds,  it  is  evident 
that  the  E^-2D^  do  not  involve  the  c$  nor  the  c$;  also  the  J$+2H™ 
involve  neither  the  cf^  nor  the  c^  ,  and  vanish  with  the  cf^  .  Further,  since 
the  xt<l(r)  and  w,,i(r)  may  be  written 


-7T  \ 


qtT  +         +  2c%COSqtT  +   2 


(61) 


it  is  found  that  the  cj\  enter  the  J$+2H$,  which  are  the  coefficients  of 
sin  qtr,  in  precisely  the  same  way  as  the  cj\  enter  the  E^—2D^,  which 
are  the  coefficients  of  sin  (9,7  -\-TT,  2). 

In  the  treatment  of  Case  II  for  Type  I  it  was  shown  that,  when  A45^0,  the 
equations  E($  —  2Dl°$  —  0  admit  a  unique  solution  for  the  cfl  0  =  1,  .  .  .  ,  k)  . 
The  same  conclusion  is  evidently  valid  here;  and  it  also  follows  at  once  that 
the  equations 

0  (i=l,  .  ..,fc)          (62) 


admit  a  unique  solution  for  the  c™  O'=l,  .  .  .  ,  A"),  namely,  c^  =  0.  In 
(62),  c^J  does  not  appear,  being  already  zero;  but  this  does  not  affect  the 
conclusion,  since  there  are  obviously  no  more  solutions  for  A:  —  1  of  the 
unknowns,  after  the  last  one  has  been  determined,  than  there  are  for  all  k. 
The  Hfx  now  vanish  identically. 

The  c"j  ,  c^  ,  cf\  are  now  determined  as  for  Type  I  ;  but  the  c"J  remain 
undetermined  until  the  next  step,  and  the  solutions  are  at  present  given  by 


PROBLEM    OF   A:   8A  I  Kl.1.1  I'ES.  H7 

(58),  as  in  Case  I  of  Type  II,  where,  however,  the  c*  and  c£,  as  well  as  tlu- 
c",',  remain  undetermined. 

In  getting  the  third-order  terms  certain  constants  will  be  determined: 
thec*  and  c£  by  the  vanishing  of  the  E^-ZD"  and  J%  +  21I;'.  rwpeo- 
tivrly;  and  the  c",'  by  the  vanishing  of  the  H™,  which  reduce  to  their  values 
in  Case  I  when  the  c"i=0. 

Likewise,  the  c™,  c"  of  any  order  are  determined  in  the  next  order, 
(u+1),  and  the  c/i  in  the  order  (»+2).  Each  set  is  obtained  from  linear 
equations,  whose  determinant  remains  the  same  for  each  successive  order. 
Thus,  with  the  same  (x>ssible  exceptions  as  in  Case  I,  it  is  impossible  in  Case 
II  to  determine  the  constants  otherwise  than  as  for  Type  I. 

!{<•  marks:  (1)  If  either  of  the  determinants  A,  or  As  vanishes  for  some 
value  of  o,  there  may  still  be  no  new  values  for  the  constants  of  integration 
which  would  satisfy  later  conditions  in  subsequent  differential  equations 
and  render  the  solutions  periodic  in  form.  Whether  the  series  converge  for 
this  value  of  a  would  be  unknown;  so  that  the  mere  vanishing  of  A,  or 
A,  would  not  warrant  the  conclusion  that  orbits  of  Type  II  exist. 

(2)  The  foregoing  conclusions  extend  beyond  a  denial  of  the  possibility 
of  obtaining  equations  of  periodic  orbits  by  a  certain  method  of  analysis, 
or  solving  the  differential  equations  in  a  certain  way:  the  non-existence  of 
a  class  of  physical  orbits  of  a  certain  type  is  asserted,  though  possibly  there 
exist  numerous  individual  orbits  of  the  type.  All  orbits,  periodic  or  not, 
arising  from  any  set  of  Ac,,  and  n  [see  (17)  and  (7)],  can  be  represented  by 
power  series  in  these  parameters  converging  through  the  interval  0^7-^2*-, 
provided  that  the  parameters  are  sufficiently  small.  And  from  the  exist- 
ence of  a  class  of  periodic  orbits  of  the  type  sought  would  follow  the  existence 
of  a  range  of  values  of  the  Ac,,  and  n  (including  zero)  satisfying  the  periodicity 
conditions.  These  equations  obviously  could  not  be  satisfied  for  a  range  of 
values  of  M  by  arbitrary  values  of  the  Ac,, ,  and  therefore  they  would  define 
the  Ac,,  as  functions  of  n,  holomorphic  for  n  sufficiently  small  and  van- 
ishing with  At.  The  substitution  of  these  values  of  the  Ac,,  into  the  original 
developments  of  the  coordinates  would  render  the  latter  series  in  M  alone, 
having  properties  (40a,  6,  d).  If  these  series  are  impossible  save  for  Type  I, 
then  there  does  not  exist  a  class  of  physical  orbits  of  Type  II  growing  out  of 
the  circles  named. 

218.  Concerning  Orbits  of  Type  IE. — It  may  be  inquired  whether  there 
r\ists  a  class  of  periodic  orbits  growing  out  of  circular  orbits  of  an  infini- 
tesimal system  which  has  no  "grand  conjunctions";  that  is,  whether  then- 
are  periodic  solutions  of  (7)  when  some  of  the  X,  have  other  values  than  are 
possible  in  Type  I. 

Let  the  initial  conditions  be  (17),  let  some  instant  when  A/,  is  at  an 
apse  be  selected  as  the  origin  of  time,  and  let  the  origin  of  longitude  be  the 


448 


PERIODIC   ORBITS. 


apsidal  position  of  Mt  ,  both  for  ^  =  0  and  for  /z^O.*  Then  Ah  =  0,  and 
w  =  T  =  0  identically  in  k  and  /x;  and  the  conditions  for  periodicity  are  (13). 
But  equations  (7)  admit  two  integrals,  an  examination  of  which  shows  that 
two  equations  of  (13),  namely  xk(2ii)  =xt(0)  and  x't(2ir)  =  x't(Q)  ,  are  a  conse- 
quence of  the  other  4k  —  2  equations.  Hence  equations  (13)  become 


(a) 


(6) 


(c) 


(d) 


0  =  ef(l  -  cos2qiir)  +M  Xli0-\-etrt(  -  sin  2<7,ir) 


-cos2gi7r) 


=  e,(g(sin 


0  =  et  (  -  2qtT^  cos  2^)  +  M  I  F,',0  +  6,7,.  (2^  sin  29i7r 


/=! 


(t  =  l,  ...  .  ,*-!), 


(63) 


where 


and  the 
here. 


(j=l,  3,  4),  etc.,  are  constants  whose  values  will  not  be  needed 


*In  treating  orbits  of  Type  I.  the  instant  of  a  symmetrical  conjunction  was  regarded  as  the  beginning  of 
a  period  and  was  taken  as  the  origin  of  time;  but  this  was  merely  for  simplicity,  since  in  periodic  motion 
any  other  instant  could  be  so  regarded.  By  taking  as  T  =0  the  instant  when  Mt  of  the  infinitesimal  system 
is  at  arbitrarily  selected  longitude,  and  choosing  that  longitude  as  a  new  origin  of  longitude  (so  that  Xt  =  0), 
it  is  clear  that  the  longitudes  which  the  other  infinitesimal  satellites  have  at  that  instant  constitute  a  set 
of  X's,  not  all  of  which  are  multiples  of  ir.  Each  family  of  orbits  of  Type  I  may  thus  be  said  to  arise  from 
any  one  of  an  infinitude  of  sets  of  the  \t  other  than  multiples  of  w,  though  all  the  sets  have  X*  =  0.  By  reason 
of  the  wt  (T),  which  are  in  general  distinct  from  zero,  the  absolute  longitude  t><  of  any  finite  satellite  at  the 
new  r  =  0  would  vary  with  MJ  (but,  since  families  of  orbits  of  Type  I  exist  for  every  position  of  the  line  of 
conjunction,  one  may  obtain  a  family  in  which  the  initial  longitude  of  Jl/t  is  zero,  identically  as  to  n,  by 
selecting  orbits  for  different  n's  from  different  families,  taking  the  conjunction  line  as  needed  for  the  M  used.) 
For  such  an  origin  of  tune  M*  would  not  in  general  be  at  an  apse  for  T  =  0. 

Hence  if  the  origins  of  time  and  longitude  be  chosen  at  an  apsidal  position  of  Aft  for  all  values  of  M,  the 
sets  of  X,  other  than  multiples  of  r,  which  can  give  rise  to  orbits  of  Type  I,  are  largely  excluded.  (Whether 
any  such  sets  remain,  depends  upon  whether,  in  Type  I,  Mt  has  any  apses  other  than  at  the  symmetrical 
conjunctions  of  the  system;  and  this  has  not  been  ascertained.) 


riO>BI.KM    (IK    /,     >  \TKI.I. I  IKS. 

While  the  determinant  of  the  linear  terms  of  tin-  A;/  .  -  .  u.-,,  and  T,  in 
ii;:<)  is  zero.  yet.  when  </,  is  not  an  integer,  solutions  of  (63)  (c)  and 
exist  for  the  An,  and  c,  as  power  -eries  in  n  and  the  u,  and  T,.  Prop- 
erties of  the  series  «i:{,.  similar  to  (19).  are  racily  established,  which  show 
that  the  solutions  for  the  An,  and  <•,,  and  also  the  scries  obtained  by  sub- 
stitution of  these  solutions  into  (63)  (a)  and  (6),  carry  n  as  a  factor.  After 
this  substitution  and  a  division  by  n,  (63)  (a)  and  (6)  become 


..o     2(1       i* 

/• 


sillier        w>u-»  rv'(4»,          8ln2g.y         ur'Mnl 

-COS27.T)  *•*  J  H'T/L  A<  /  T  2(1  -C082«7(ir)  W  '•'  J  / 

+T 

'- 


ifi4) 


The  constant  term  in  (64a)  vanishes,  since  for  qt  not  an  integer  no  relation 
(22)  holds;  the  constant  term  in  (646)  reduces  to  -2qtCIJt.  If  any  of  the 
CtM  are  distinct  from  zero,  (64)  are  not  satisfied  by  w,=-T,  =  p-0;  hence 
wt  and  T  do  not  exist  as  holomor])hic  functions  of  /*  vanishing  with  n.  or 
periodic  orbits  of  the  type  sought  do  not  exist.  The  necessary  condition 
for  periodicity,  namely,  that  the  CIJt  vanish,  is,  by  (28), 

2'd,,  2  e.,(«>inw(X,-X,)=0         (i-1  .....  k).      (65) 

j         m-l 

Of  these  A-ccjuations  in  the  quantities  XyO'»  1,  .  .  .  ,  k—  1),  one  is  evidently 
a  consequence  of  the  others;  for,  before  Xt  is  put  equal  to  zero,  the  Jacobian 
of  the  CtJlt  with  respect  to  the  X;(t  =  1,  .  .  .  k;  j=  1,  .  .  .  ,  k)  is  identically 
zero.  Let  the  equation  for  t  =  1  be  suppressed. 

If  particular  values  be  assigned  to  all,  save  one,  of  the  X;  ,  the  last  X  can  still 
be  given  an  infinitude  of  values  for  which  any  one  TM  is  distinct  from  zero: 
hence  equations  (65)  imixw  very  special  conditions  upon  the  X,.  Whether 
there  are  any  sets  of  X's,  other  than  those  of  Type  I  which  satisfy  (65),  is 
unknown.*  It  will,  however,  be  shown  that  there  are  no  others  "in  the 
vicinity  of"  multiples  of  T,  provided  a  is  sufficiently  small. 

•TV  limitation*  upon  any  »uch  net*  «r*m  fully  a*  «evere  here  aa  in  Type  I. 


450  PERIODIC    ORBITS. 

If  the  C1|0  are  developed  as  power  series  in  the  \t—J,ir  («/,  =  (),  1),  the 
coefficients  of  the  linear  terms  are  simply  [dC,,0/d\f]  for  \j  =  Jjir.  Evidently, 
from  (28), 


_ 


m-i 


(66) 


Denoting  by  S{/  the  coefficient  of  X,  in  the  itt  equation  of  (65),  and  intro- 
ducing a  by  (35),  the  Stl  are  obtained  as  power  series  in  a.  It  is  found 
that  the  lowest  exponent  of  a  present  in  S(f  is  (/—  i)  if  f  <i  and  is  5(/  —  i)  if 
f>i.  Hence,  of  all  the  Stf  (f  <i),  Stl  carries  the  lowest  power  of  a,  namely, 
the  (1— t)th;  and  of  all  the  Stf(f>i),  Sttt+1  carries  the  lowest,  namely,  the 
fifth.  Consequently,  if  a'1"'1  be  removed  as  a  factor  from  all  the  Su 
'(/=!,  .  .  .  ,  k—  1;  i  =  2,  .  .  .  ,  k),  a  determinant  A7  is  obtained  (equal  to 
the  determinant  of  the  coefficients  Sif  multiplied  by  a  power  of  a) ,  in  whose 
r*A  row  all  elements  save  those  of  the  first  and  (r-\-l)th  columns  begin  with 
a  power  of  a.  Therefore  A7  is  of  precisely  the  same  type  as  A, ,  and  the  dis- 
cussion of  the  latter  shows  also  that  A7  (and  hence  the  determinant  of  the 
St/)  is  distinct  from  zero  for  all  values  of  a  sufficiently  small.  Therefore 
quantities  A/(/  =  l,  .  .  .  ,  fc— 1)  exist  such  that  equations  (65)  are  not 
satisfied  for  any  values  of  the  X,  for  which  \\f—Jfir\  <A,  except  X/=0  or  IT. 

Moreover,  the  existence  of  a  set  of  X/  satisfying  (65)  would  not  prove  the 
existence  of  w,  and  r,  as  functions  of  M  satisfying  (64).  If  it  is  assumed  for 
the  moment  that  orbits  of  Type  III  exist,  and  the  method  of  construction  is 
examined,  equations  related  to  (65)  are  encountered.  Thus,  in  finding  the 
second-order  terms,  the  //",  must  be  made  to  vanish,  and  each  of  these  involves 
Fourier  series  in  the  various  X,  —  X, .  When  qk  is  an  integer,  somewhat  differ- 
ent difficulties  arise  in  the  existence  proof;  the  same  difficulty  is,  however, 
encountered  in  the  construction. 

A  general  negative  conclusion  is  not  yet  warranted;  but  it  is  evident 
that  if  any  orbits  of  Type  III  exist,  they  must  satisfy  very  special  conditions. 
In  every  case,  however,  periodic  orbits  of  the  type  sought  do  not  exist  if 
(22)  holds. 

219.  Concerning  Lacunary  Spaces. — The  relation  (22)  may  be  expressed 
in  the  form  J(pf—pt')=p,+qk',  hence  this  relation  can  hold  only  if  qt  is  an 
integer.  But  the  converse  holds  true  only  when  k  =  2.  For  example,  in 
the  following  selections  qt  is  an  integer,  but  no  relation  (22)  holds: 


PHOHI.K.M    UK    /.'    >.\  I  KI.l.I  I  K.-.  }.',i 

When  k  =  2.   (22)  always  holds*  if  qt  is  an  integer;  for  then  ^=1  (other- 
wise T  would  not  l.c  the  smallest  synodic  period  of  the  infinitesimal 
and  hence  Jp{  =  qt  is  satisfied  by  giving  J  the  integral  value  qt. 

Since  /?,7'  =  2<ytT,  the  case  <?,  an  integer  is  the  one  \\here  the  consecutive 
conjunct  ions  of  the  infinitesimal  system  occur  at  the  same  absolute  longitude; 
Utd,  denoting  the  synodic  period  of  the  two  infinitesimal  satellites  M,  and  .17, 
1>\  Tfl,  since 

"Jr.  =  ».    2r     ' 
"/-"» 

it  follows  that  »,7'/f  =  2ir./  when  (22)  holds;  or  the  consecutive  conjunc- 
tions of  the  infinitesimal  pair  M,  and  M,  occur  all  in  the  same  absolute 
longitude. 

Moreover,  since  all  the  u\  vanish  at  the  beginning  and  end  of  each  period, 
all  the  "grand  conjunctions"  of  the  finite  system  occur  at  the  same  longi- 
tudes as  those  of  the  infinitesimal  system,  and  intermediate  conjunctions 
of  any  finite  pair  occur  at  very  nearly  the  same  longitudes  as  those  of  the 
corresponding  infinitesimal  pair. 

These  facts  suggest  a  physical  reason  for  the  non-existence  of  periodic 
orbits  under  certain  circumstances.  The  greater  part  of  the  mutual  dis- 
turbances of  two  bodies  occur  while  they  are  near  conjunction;  and,  if  the 
consecutive  grand  conjunctions  occur  at  exactly  the  same  longitude,  the 
perturbations  of  the  elements  would  tend  to  be  cumulative.  Nevertheless, 
if  there  are  more  than  two  bodies,  the  mutual  disturbances  may  so  balance 
each  other  as  to  yield  periodic  orbits,  especially  if  the  bodies  are  far  apart 
(t.  e.,  a  sufficiently  small)  unless  (22)  holds.  But  if  two  bodies  have  con- 
junctions between  the  grand  conjunctions,  all  occurring  very  near  the  same 
longitude,  the  other  bodies  can  not  counterbalance  the  large  perturbations  of 
the  two. 

More  exactly,  there  exists  a  range  of  values  of  the  masses  and  the  et, 
including  zero,  for  which  periodic  orbits  are  impossible;  so  that,  unless  the 
orbits  for  n  =  0  are  eccentric  rather  than  circular,  there  are  for  small  values  of 
/u  no  periodic  plane  orbits  of  k  satellites  when  (22)  holds. 

So  far  as  this  result  extends,  it  would  indicate  that  no  asteroids  having 
nearly  circular  orbits  would  be  found,  whose  periods  compared  to  Jupiter's 
are  in  the  ratios  J,  f  ,  f  ,  etc.  Those  whose  periods  are  nearly  in  any  such 
ratio  should  be  found  subject  to  very  great  perturbations. 

It  is  well  known  that  lacunary  spaces  of  the  sort  just  mentioned  do 
occur  among  the  asteroids.  That  there  are  such  spaces  also  when  the  ratio 
of  the  periods  is  J,  and  other  such  values,  is  not  surprising,  as  in  any  case 
the  slightest  deviation  from  the  correct  initial  values  destroys  periodicity, 
there  being  Poisson  terms  in  the  solutions. 


•In  PoinrmvV  .li*-UH.«iMii  of   Ih.-  |.rol,l.-in   ..f   thr.-.-  I..,  I,-     i|,.-r.|..n      <\,'-   MM    •hw     ,,    ,-    • 
without  such  a  relation  aa  (22)  holding  does  not  ari*e. 


452  PERIODIC    ORBITS. 

220.  Jupiter's  Satellites  I,  II,  and  HI. — Of  Jupiter's  longer  known  satel- 
lites, the  innermost  three  move  almost  exactly  in  a  plane,  apparently  in 
periodic  orbits  having  symmetrical  conjunctions;  and  their  masses  with 
respect  to  that  of  the  planet  are  very  small.  Since  for  orbits  of  Type  I  the 
increase  in  the  longitude  of  Mt  during  a  period  is  independent  of  //,  being 
equal  to  ntT,  the  average  angular  velocity  of  each  finite  satellite  for  a  period 
may  be  taken  as  the  corresponding  n. 

The  unit  of  time  being  the  sidereal  day,  and  the  unit  of  mass  being  the 
mass  of  Jupiter,  the  observational  data  are  :* 


,  =  0.000017,  Mt  =  0.000023,  Mt  =  0.000088, 

n,  =  3. 551552261,  n,= 1.769322711,  n,=  .878207937. 


(67) 


Since  (w1-n,)/3  =  . 891 114775,  and  w,-n,  =  . 891 114774,  the  w,  of  (67)  satisfy, 
far  beyond  observational  accuracy,  the  equations  (2),  where  v  =  .891114774. 
Then  the  period  T  is  7.0509271  days,  and 

Pi  =  3,  pt  =  l,  q,  =  .985516077,1 

(68) 

<t>K  =  T,  } 

so  that  satellite  III  advances  354?785788  during  each  period,  while  satel- 
lites I  and  II  advance  1080°  and  360°  respectively  more]  than  this.  If  /z 
is  taken  arbitrarily  as  .0001,  then 

ft  =  .17,  ft  =  .23,  ft  =  .88. 

It  seems  desirable,  however,  to  retain  the  /8's  in  the  computations,  inasmuch 
as  a  new  determination  of  the  masses  may  render  it  necessary  to  use  other 
values  than  those  given  above. 

The  D™  and  E™  of  (46)  are  obtained  by  writing  equations  (43)  in  the 
form 


(69) 


1  m-0 

where,  rearranging  according  to  multiples  of  T, 


*Tisserand,  Traite  de  Mfcanique  Celeste,  vol.  4,  p.  2.     The  nt  Riven  there  (203?48895528,  101?37472396, 
and  50T31760833)  are  here  reduced  to  circular  measure. 


PROBLEM    OF   tc   SATKU.1TE8. 

Evidently  the  U™.  V™,  and  U£  enter  respectively  the  D™,  D™.  and  D%, 
etc.  From  (23),  (5), and  the  relation  (n,/n,Y=(n./n,Y.  the  «(/and  S,,are  found 
to  be 


€„  =  «„  =  .  6284333,        «,,  =  «„  =  .  3939606,  «„-««  =  .  6268932, 

*»?»  =3.1366  A,          *»,*''  =  1.23875/3,,  ^  =.19132/3,, 

«u  nil  =  !  2326  ft  ,  =  .77464  /3,  ,  =  .30443  /3,  . 


,7(1 


The  Fm(e,j),  and  also  the  Gm(etJ),  etc.,  encountered  in  the  higher  orders  are 
readily  computed  by  using  the  tables  of  coefficients  given  by  LeVerrier.* 

In  obtaining  the  successive  A™  and  B™  from  (52),  the  smallest  divisors 
introduced  are  16-tf,4-gJ,  and  1-gJ,  or.  1156616,  .05772591,  and  .02875806 
respectively.  These  divisors  decrease  materially  the  effectiveness  of  the 
small  value  of  /*;  nevertheless  the  terms  above  those  of  the  second  order 
seem  relatively  unimportant  and  will  not  be  computed.  The  coefficient^ 
of  n  in  the  X,(T)  and  w,(r)  are  found  to  be:f 


*U(T)  =  (.3/3,  -  .1/3,)  -  .8/3,cos2T  -  .l&cosSr  -  203.70scos4r 
+  (.7/31-.2/31)cos6T-.2/3tcos8T+.lftcoslOr+ 


-  (1.1/3,-.  3/3,)sin6r  +  .3/31sin8T-3/3Jsin9T-.l/31sinlOr  + 

T,.,(T)  =  (.5/3,  +  .3&-  .1/3.)  +  .8/3,  COST-  (58.4/3,  +  100.1^,)  cos2T  -  .7/3,cos3r 
(.7/3,-.2/31)cos4T-.l/3Jcos5T-.2/3,co86T+.l/JIcos8T  + 


-(.8/3,-.2/3,)8in4r+.l/3Jsin5T+.2/31sin6T-.l/3,sin8T 


(.4/3,  +  .5/3,+  .3&)  +28.9/3.COST+  .6/3,cos2r  -  (.4/3,  -  .2ft)  cos.'lr 
+  .l/3,cos4T  •  •  •  , 


For  the  second-order  terms  the  £>!!»'  and  E™  are  found  from  (54),  where  it 
is  convenient  first  to  rearrange  the  coefficients  of  <r^i  and  a~J,  according  to 
multiples  of  T.  The  xfj  and  io/4  involve  all  multiples  of  T;  but,  as  they 


MimaiM  <fc  rOb«rM(otr«  «fc  Pom  (Jtf*m»irM)  Tol.  2.     SuppUanent. 

fThe  writer  regret*  to  •tlkte  t*iat  l^e  va'ue>  formerly  published  (rranMe(ion4  </  <A<  Xmmean  .Malkt- 
malical  Society,  vol.  9,  pp.  29-33)  are  practically  all  erroneous,  the  factor  g,  of  the  la*t  (mm  of  (46)  having 
been  overlooked  in  making  the  calculation. 


454  PERIODIC   ORBITS. 

carry  the  factor  /?>  all  save  the  following  terms  fall  below  the  limit  of  accu- 
racy in  the  xtil  and  wl:l  above.  To  facilitate  comparison,  the  second-order 
terms  are  shown  multiplied  by  n  —  .0001.  They  are 


-(2.9/31/32-12.2$-9.2&/33)cos4T-2.0/32!cos8T  + 


.0001  WU(T)  =  .Sftftsinr-  (.3/31&  +  .2$+.3&&)sin2r  +  (5.8/3^  -24.  4ft2 
-18.3/32/33)sin4r+.l&&sin6T+5.1/322sin8T  •  •  •  , 

.0001z2.2(r)  =  (.2/3I+.6&&+.5/3D  -  (.!&&+.  1$)  COST 

+  (6.3$+12.6/31/32+12.6&/33+3.1$)cos2T 


.0001  wM(T)  =  (.2j81/3,+.2|81/3l 

+  1.3ft&+6.2/33Osin2T  +  (.l/332sin3r  +  .4#  +  .  1  ft  &+  1.40,0, 
+  1.30J)sin4r+ 

.0001  XM(T)  =  -  (2.3ftft+2.5$+3.9ftft)  COST-.  I^cos2r 
-.Iftftcos7r+  •  •  •  , 

)  =  (4.7ftft+4.9/32!+7.6ftft)sinT+.2ft2sin2H  ----  . 


Since  /•(  =  a((l+z(.,M+z,,2M2+  •  •  •)  and  vf  = 
the  radius  vector  and  absolute  longitude  of  each  satellite  are  obtained  by 
computing  the  a,  from  (3)  and  using  the  coefficients  above;  the  deviations 
from  their  values  in  the  undisturbed  circular  orbits  are  given  to  five  significant 
figures,  so  far  as  the  terms  of  the  first  two  orders  are  concerned,  How  much 
these  would  be  affected  by  terms  of  higher  orders  is  unknown;  in  fact  no  proof 
has  been  given  that  the  series  converge  for/z  =  .0001,  although  they  have  been 
proved  convergent  for  all  /x  sufficiently  small. 

To  show  the  general  shape  of  these  orbits,  the  values  of  the  vt  and  r(/at 
will  be  given  to  four  decimals,  using  the  values  of  the  ft  tabulated  above: 

r]/a1  =  l-.0044cos4r,        w1  =  7r+3.9855T+.0089sin4r, 

r2/a,=  l-.0093cos2T,        »,=  1.9855  T-.  0003  sin  r  +  .  0185  sin  2  T 

+  .  0001  sin  3  r+.  0001  sin  4  T, 

rl/a,=  l  +  .  0006  COST,          ^  =  .9855r-.0011sinT. 


Hence  if  these  orbits  are  thought  of  as  ellipses  rotating  in  the  plane, 
the  major  semi-axes  would  be  the  respective  a,,  the  several  eccentricities 
would  be  .0044,  .0093,  .0006,  and  the  axes  would  rotate  forward  at  rates 
whose  average  values  are  the  nf.  The  three  satellites  are  in  line  with  Jupiter 
at  the  beginning  and  middle  of  each  period,  II  and  III  being  on  the  same  side 
of  the  planet  at  T  =  0,  and  I  and  III  on  the  same  side  at  T  =  TT.  Whenever 


PROBLEM    OF   k   SATELLITES.  455 

II  is  in  conjunction  with  I  or  III,  the  inner  of  the  pair  is  near  a  perijove  and 
the  outer  is  near  an  apojove,  which  decreases  the  amount  of  their  mutual 
perturbations. 

No  radius  vector  or  longitude  differs  very  widely  at  any  time  from  its 
value  in  a  circular  orbit  (a,  and  \+qj,  respectively).  The  largest  departures 
arc  for  satellite  II,  as  fj/o,  reaches  a  minimum  of  .9907  at  r  =  0  and 
a  maximum  of  1.0093  at  about  r  =  r/2  and  every  half-period  thereafter, 
t',  meanwhile  ranging  from  64'  more  to  64'  less  than  the  mean  longitude  of  II. 
Similarly,  satellites  I  and  III  get  30'  and  4'  respectively  ahead  of  and  behind 
their  mean  positions,  and  the  r,/o,  at  such  instants  closely  approximate  their 
mean  value,  unity.  For  satellite  I  the  maxima  of  r/o  occur  at  intervals  of 
a  quarter-period,  and  for  satellite  III  they  occur  at  intervals  of  a  period. 

Finally,  it  may  be  noted  that,  for  this  system  of  bodies,  the  increments 
Acu  (see  §  209)  which  have  been  given  to  the  initial  values  of  the  coordinates 
and  their  time-derivatives  to  preserve  periodicity  when  the  bodies  are  finite 
are  approximately 

r<  <  v,  v, 

-.0044  a,  0  0  .0310  (radians  per  day) 

-.00930,  0  0  .0329  (radians  per  day) 

+  .0006o,  0  0  -.0010  (radians  per  day) 

221.  Orbits  About  an  Oblate  Central  Body.*  —  If  the  central  body  is  an 
oblate  spheroid  and  the  satellites  are  spheres  moving  in  its  equatorial  plane, 
periodic  orbits  of  Type  I  still  exist,  the  successive  grand  conjunctions  falling 
at  the  same  or  a  different  longitude,  according  as  qt  is  or  is  not  an  integer. 

The  differential  equations  for  this  case  are  obtained  from  (4)  by  simply 
multiplying  each  1/rJ  (j  =  1,  .  .  .  ,  k)  by  /(r,),  where 


o  being  the  equatorial  radius  and  e  the  eccentricity  of  the  spheroid.f 
Let  oV/aJ  =  7,M;  then  equations  (7)  are  replaced  by 

(a)     xri'+2x[(w'l+ql)+»2'6lsri8m(<t>),+wj-wl) 


•ThwMction  take*  a  firrt  step  in  »  direction  Mggerted  by  Professor  K.  Lave*,  particularly  with  rafenoM 
to  Jupiter's  satellite*. 

tMoulton,  Cele*ial  Mechanic*,  p.  122,  where  a'+r-rj,  and  2«fc'v'i-..-3t'Af/2a«. 


456  PERIODIC    ORBITS. 

This  substitution  requires  the  flattening  to  vanish  with  the  masses 
Af  ,,...,  Mt  ,  so  that  the  central  body  becomes  spherical  if  the  others 
become  infinitesimal;  but  the  amount  of  flattening  corresponding  to  any 
given  set  of  finite  masses  remains  quite  arbitrary,  even  if  the  pt  and  qt  are 
specified;  for  the  values  of  the  a,  merely  determine  the  ratios  of  the  7  , 
and  one  7  may  be  taken  at  pleasure.  In  the  solutions  of  (71)  satisfying 
initial  conditions  (17),  the  terms  independent  of  ju  are  the  same  as  formerly. 
Hence  in  Case  I  where  qt  is  not  integral,  the  Aw,  and  et  still  exist  as  convergent 
power  series  in  n  satisfying  (29  a,  6),  though  of  course  their  values  in  terms 
of  n  are  now  different  because  of  the  7,  .  Thus  periodic  orbits  exist. 

In  case  qt  is  an  integer  the  j,  enter  the  xf(0;  T)  and  x,(i;  T),  but  do  not 
appear  in  the  Ptl  or  P(f  of  A3.  Thus  the  argument  in  Case  II  is  likewise 
unaltered,  and  periodic  orbits  of  Type  I  exist  under  the  same  conditions 
as  when  the  bodies  are  all  spherical. 

The  numerical  results  for  Jupiter's  satellites  given  above  are  affected 
but  slightly  in  the  first  and  second  orders  by  including  the  flattening  of 
Jupiter.  The  corrections  to  be  added  are  in  fact 

to  o:M  (T)       add        .17,  cos  4r 

to  .0001  a-,,,  add      1.8  #,7,  cos  4r 

to  .0001  WM  add  -3.6  &7,  sin  4r 

to  z,,i(T)        add        .l72  cos  2r 

to  .0001  xta  add      (.2/3,+.  4ft)  72  cos  2r 

to  .0001  My,  add  -(.50I  +  .8ft)71  sin  2r 

toar,,,(T)        add       .17,  cos  T 

to  .0001.T,,,   add  -     .1&7,  cos  r 

to  .0001  w,iZ  add        -1&7,  sni  r 


And  since  7,  =  .36,  72  =  14.3,  and  73  =  5.7,  the  values  of  the  rt/at  and  vt 
given  above  are  changed  as  follows: 

to  r,  a,  add  .0018  cos  4r.  to  r2/a2  add  .0007  cos  2r, 

to    r,     add  -.0030  sin  4r,  to    v2    add  -.0011  sin  2r; 
then 

r,  a,  =  1  -  .0026  cos  4r,  t;,  =  Tr  +  3.99855r+.0059  sin  4r, 

r,/a2=  1  -  .0086  cos  2r,  v,=  1.9855r-  .0003  sinr  +  .0174  sin  2r 

+  .0001  sin  3r+.0001  sin  4r, 

r3/a3  --=  1  +  .0006  cos  T,  v3=  .9855r  -  .001  1  sin  T. 


CHAPTER  XV. 

CLOSED  ORBITS  OF  EJECTION  AND  RELATED 
PERIODIC  ORBITS. 


222.  Introduction. — In  the  problem  of  two  bodies  there  is  in  no  sense 
( •ontinuity  between  circular  orbits  revolving  in  the  forward  and  retrograde 
directions,  except  where  their  dimensions  shrink  to  zero  or  become  infinitely 
great.     But  in  the  restricted  problem  of  three  bodies  the  deviations  from 
the  circular  forms  of  the  orbits  are  such  that  there  is  geometrical  continuity 
in  some  classes  between  those  which  revolve  in  the  forward  direction  and 
those  which  are  retrograde;  and  the  limit  between  the  two  types  is  an  orbit 
passing  through  one  of  the  finite  bodies.     If  the  infinitesimal  body  leaves 
one  of  the  finite  bodies,  its  orbit  is  called  an  orbit  of  ejection;  and  if  it  strikes 
a  finite  mass,  it  is  called  an  orbit  of  collision. 

In  certain  cases  orbits  of  ejection  are  also  orbits  of  collision,  or  closed 
orbits  of  ejection.  When  the  direction  of  collision  is  exactly  opposite  to 
that  of  ejection,  they  are  the  limits  of  two  classes  of  periodic  orbits,  in  one 
of  which  the  motion  is  direct  and  in  the  other  of  which  it  is  retrograde. 
The  closed  orbits  of  ejection  are  not  themselves  periodic  orbits,  even  if  the 
physical  impossibility  be  disregarded  and  the  problem  considered  purely 
from  the  mathematical  point  of  view;  for,  if  the  expressions  for  the  coordi- 
nates are  followed,  in  the  sense  of  analytic  continuity,  beyond  the  values  of 
t  for  which  a  collision  occurs  they  become  complex,  and  never  become  real 
again  for  increasing  real  values  of  /.  Those  orbits  in  which  the  ejection  and 
collision  are  not  in  opposite  directions  are  not  the  limits  of  periodic  orbits, 
or  at  least  of  orbits  which  re-enter  after  a  single  revolution. 

The  object  of  the  investigations  of  this  chapter  is  to  determine  the 
limiting  types  of  certain  classes  of  periodic  orbits,  and  thus  partially  to 
prepare  the  way  for  the  discussion  of  the  evolution  of  the  various  classes  of 
[>oriodic  orbits  with  varying  values  of  the  parameters  on  which  they  depend, 
and  to  show  the  relations  among  these  various  classes.  The  existence  of 
the  closed  orbits  of  ejection  will  be  established,  some  of  their  properties  will 
be  derived,  and  it  will  be  proved  that  each  one  in  which  the  direction  of  ejec- 
tion and  collision  is  opposite  is  the  limit  of  two  series  of  periodic  orbits. 

223.  Ejectional  Orbits  in  the  Two-Body  Problem.     As  preliminary  to 
the  general  problem,  the  special  case  in  which  there  is  only  one  finite  mass 
will  first  be  treated.     Let  the  mass  of  the  finite  body  be  1  - n  and  let  the  units 

to  chosen  that  the  gravitational  constant  is  unity.     Then  the  motion  of 

457 


458  PERIODIC    ORBITS. 

the  infinitesimal  body  projected  along  the  fixed  £-axis  satisfies  the  differential 
equation  d?£  _  _  i-M 

df  ="  +  T 

where  the  sign  is  —  or  +  according  as  the  motion  is  in  the  positive  or  negative 
direction  from  the  origin. 

Suppose  f  =  f0  and  d£/dt  =  £'  =  £0'  at  T  =  TO  .      Then  the  first  integral  of 
equation  (1)  becomes 

#V  _,./«_  +  2(1-M)  -  2  (l-/i)    ,   r,2_+2(l-M)    ,  m 

di)  1         "IT  ~F      '• 

If  Cj  is  negative,  |£|  has  a  finite  maximum  for  which  £'  vanishes;  if  c^  is  zero, 
%  approaches  zero  as  |£|  becomes  infinitely  large;  if  ct  is  positive,  %  is  finite 
for  |£|  infinite.  It  will  be  assumed  that  c,  is  negative  in  order  to  get  orbits 
of  ejection  which  are  closed.  Then,  without  loss  of  generality,  it  can  be 
supposed  that  £0  is  the  greatest  value  of  £  for  projection  in  the  positive  direc- 
tion, or  the  least  for  projection  in  the  negative  direction.  Then 

€1-0,        Cl=T^P^-  (3) 

?o 

With  the  initial  values  (3),  the  integral  (2)  becomes 


where  j  is  an  integer. 

Now  consider  £  as  a  function  of  (t  —  ta).  Since  the  right  members  of 
(1)  and  (2)  are  regular  for  all  values  of  t  and  all  values  of  £  except  £  =  0, 
it  follows  that  £  is  a  regular  function  of  t  for  all  values  of  t  except  those  for 
which  £  vanishes.  These  values  of  t  are  easily  determined  from  (4),  and  are 
found  to  be 


where  j  takes  all  integral  values. 

The  character  of  $  as  a  function  of  (t  —  0  in  the  vicinity  oit  =  t,  is  easily 
determined  from  (4).     The  left  member  is  expansible  as  a  power  series  in 

=  11,  and  the  equation  can  be  written  in  the  form 


It  is  found  that 


,  ,  ,  . 

Therefore  17  is  expansible  as  a  power  series  in  (t—  tj)lf3,  starting  with  a  term 
of  the  first  degree  in  (t  —  ^)1/3.  Since  ±£  =  r?2,  it  follows  that  £  is  expansible 
as  a  power  series  in  (t  —  £/)1/3,  starting  with  a  term  of  the  second  degree  in 
(t  —  t,)l/3.  It  is  easily  seen  from  (4)  that  F(ry)  is  an  odd  function  of  r?.  There- 
fore 17  is  an  odd  series  in  (t—  ^)1/3,  and  £  is  an  even  series  in  (t  —  tj)l/3.  Since 
the  only  singularities  are  given  by  (5),  the  radius  of  the  circle  of  conver- 

gence for  the  series  for  both  T?  and  £  is  7r|£0U    .~J°  .  . 


CLOSED    ORBITS    OK    KJK(TK>X    AND    HKI..V1  Kl>    I'KltlnDIC   ORBITS.          459 

The  form  of  the  solution  in  the  vicinity  of  t  =  t,  beinn  known,  the  co- 
efficients of  the  series  can  easily  be  found  from  (1)  by  the  method  of 
undetermined  coefficients.  It  is  convenient  in  the  computation  to  let 

T-tf-W*,  (6) 

nftor  which  (1)  becomes 

*  CD 


The  solution  of  this  equation  with  initial  value  of  £  equal  to  zero  has  the  form 
±{-o,T»+o4T4+  •  •  •  +O^T"+   •  •      •  (8) 

By  direct  substitution  and  comparison  of  coefficients,  it  is  found  that 


(9) 


More  convenient  formulas  for  use  can  be  developed  by  eliminating  the 
term  in  £"'  from  (7).     After  the  transformation  (6)  the  integral  (2)  becomes 


C-g  -|l/l 

=-(l—  M),  a  =  arbitrary  constant. 


On  using  this  equation  to  eliminate  £~*  from  (7),  the  result  is  found  to  be 

T£  j  »~  2£ ^  +^  ( -p)  "=+9(1— M)T*.  (10) 

Now  it  follows  from  equations  (8)  and  (9)  that 


J-t 


_  2^  =  -  4<V  fl  +  3aT«+2aV+  TJ  fj+2)  0*1* 
dr  L 


+4ar'+4aV+2 


400  PERIODIC    ORBITS. 

Hence,  on  equating  to  zero  the  coefficient  of  T*'  after  these  series  have  been 
substituted  in  (10),  it  is  found  that 

1-1 
3)o,;=  -a(2.f  +;  +  2K_2-  2  (J-k+VW-VfaA^  ,       (11) 


which  gives  the  coefficients  very  simply  for  all  values  of,/  greater  than  unity. 
The  result  of  applying  (11)  and  reducing  the  coefficients  to  the  decimal 
form  is 


+0.46996aV0-0.60863aV2+0.82861a7T14-1.16832aV"+   •  •   •]. 

So  far  as  this  solution  is  written  the  signs  of  the  terms  alternate  and 
a2>  has  a1  as  a  factor.  This  is  a  general  property  which  will  be  needed  in 
establishing  the  existence  of  the  closed  orbits  of  ejection  in  the  problem  of 
three  bodies  (§230).  It  follows  at  once  from  (11)  that  the  part  of  the  co- 
efficient a2J  which  comes  from  the  first  term  on  the  right  is  opposite  in  sign 
to  the  coefficient  of  the  preceding  term.  Since  the  sum  of  the  subscripts  of 
the  product  terms  in  the  right  member  of  (11)  is  2j,  their  product  has  the 
same  sign  as  the  coefficient  of  Oj,_2.  Therefore,  a2J  and  a2,_2  are  opposite  in 
sign  for  all  j.  It  also  follows  from  (11)  that  atj  contains  a  as  a  factor  to  one 
degree  higher  than  it  appears  in  a2J_2. 

The  expression  for  £  has  a  branch-point  at  t  =  ih  where  three  branches 
unite.  If  t  —  tj  =  pev^r'('f+*nt),  the  three  distinct  branches  are 


(12) 

If  t  —  tj  is  real  and  negative,  <p  =  ir  and  the  second  of  these  expressions  alone 
is  real.  If  t  —  t}  is  real  and  positive,  ^  =  0  and  the  first  of  them  alone  is  real. 
Consequently  the  analytic  continuation  of  the  real  branch  of  the  expression 
for  £  through  t  =  t,  leads  to  a  complex  value,  and  this  value  remains  complex 
as  t,  remaining  real,  increases  to  +  oo  .  Therefore  the  motion  is  not  strictly 
periodic. 

In  case  the  orbit  is  not  one  of  ejection  the  branch-points  of  the  functions 
which  express  the  coordinates  in  terms  of  the  time  are  not  on  the  real  axis. 
If  the  major  axis  is  kept  fixed,  and  if  the  eccentricity  is  varied  from  zero  to 
unity,  an  examination  of  the  equations  from  which  the  character  of  the 
functions  can  be  determined  shows  that  the  singularities  start  from  both 


positive  and  negative  infinity  on  the  lines  tj—t0  =  jw^f—-^Kad  approach 

the  real  axis  as  a  limit  as  e  approaches  unity.  At  these  singular  points  two 
branches  of  the  function  permute  except  when,  for  e  =  l,  two  of  them  have 
united  on  the  real  axis,  and  then  three  branches  permute.  If  e  increases 
beyond  unity,  each  of  the  branch-points  divides  into  two  which  move  equally 
in  opposite  directions  along  the  real  axis,  and  for  e  =  oo  one  of  each  pair  unites 
with  another  of  an  adjacent  pair. 


CI.OSKI)    0|<HII>    i.K     inCfKM      \M)     HKI.MKD     1'KUK  l|)IC    i.UHII-  Mil 

224.  The  Integral.  -Equation  (2)  holds  for  all  values  of  /.  and  therefore 
when  the  scries  (9)  is  substituted  in  it  the  result  is  an  identity  in  r.  The 
conditions  that  the  coefficients  of  tin-  various  powers  of  T  shall  he  identical 
in  the  left  and  right  members  furnish  severe  tests  of  the  accuracy  of  the  com- 
putation of  (9).  The  integral  also  gives  the  relation  between  the  arbitrary 
a  and  the  greatest  distance  £0.  By  direct  substitution,  it  is  found  that 


The  interval  from  ejection  to  collision  is  found  from  equation*  (5),  (9), 
and  (13)  to  be 

_r\'« 

(14) 


V2(1-M)       V2(1-M) 
For  a  =  0,  which  corresponds  to  a  parabolic  orbit,  P  is  infinite. 

225.  Orbits  of  Ejection  in  Rotating  Axes.— In  the  problem  of  three 
bodies  the  motion  of  the  infinitesimal  body  will  be  referred  to  rotating  axes. 
In  the  demonstration  of  the  existence  of  closed  orbits  of  ejection  in  the  prob- 
lem of  three  bodies  it  will  be  necessary  to  use  some  of  the  properties  of  the 
orbits  of  ejection  in  that  of  two  bodies.  For  this  reason  the  orbits  now  under 
consideration  will  be  referred  to  axes  rotating  uniformly  with  the  period  2r. 

Suppose  the  ejection  takes  place  at  t  =  tt  and  along  the  x-axis,  where  the 
rectangular  coordinates  are  denoted  by  x  and  y.  Then  x  and  y  are  given 
by  the  equations 

x=-Kcos(<-{,),        y=  -£sin  (<-/,).  (15) 


Those  orbits  which  re-enter  in  the  direction  opposite  to  that  of  ejection 
are  of  greatest  interest  in  the  present  connection.  The  condition  that  they 
shall  have  this  property  is  that  their  period  in  t  from  ejection  to  collision 
shah1  be  a  multiple  of  2*.  This  condition  becomes,  by  virtue  of  (14), 

(±  €.)M  -2;V2TI^),  (16) 


where  .;  is  a  positive  integer. 

Figs.  15,  16,  and  17  show  the  curves  for  j  equal  to  1,  2,  and  3,  at  least  as 
to  general  form,  in  full  lines  for  ejection  along  the  x-axis  in  the  positive  direc- 
tion, and  in  dotted  lines  for  ejection  in  the  negative  direction.  These  curves 
for  the  three  values  of.;'  are  not  drawn  to  the  same  scale,  for  it  follows  from  (  16) 
that  their  linear  dimensions  are  proportional  iof1*.  One  of  the  important 
properties  of  all  these  curves  is  that  they  intersect  the  x-axis  perpendicularly 
at  their  mid-points. 


462 


PERIODIC    ORBITS. 


226.  Ejectional  Orbits  in  the  Problem  of  Three  Bodies.— The  differential 
equations  of  motion  for  the  infinitesimal  body  when  the  finite  masses  describe 
circular  orbits  are,  in  canonical  units, 


(17) 


c.      = 
df"      dt      dx' 


_ 

dt      dy 


These  equations  have  the  integral 


(18) 

When  /x  is  zero  the  problem  reduces  to  that  of  two  bodies,  which  was 
treated  in  §223.     The  singularity  in  the  solution,  whether  /x  is  zero  or  not, 
comes  from  the  fact  that  i\  tends  toward  zero  as  a  limit  as  t  tends  toward 
tt .     It  is  intuitionally  clear  that 
the  mass  /x   will   have   only   a 
slight   influence   on  the  motion 
of  the  infinitesimal  body  while  rl 
is  small,  and  it  seems  probable, 
therefore,  that  the  nature  of  the 
singularity  at  t  =  t^  is  the  same 
whether  /x  is  distinct  from  zero 
or  not.     This  is,  indeed,  the  case, 
as  was  first  proved  by  Levi-Civita 
in  a  very  important  memoir.* 

It  follows  from  (18)  that  x'2 
+  7/'2  tends  toward  infinity  as  rl 
tends  toward  zero,  but  that  the 
limit  of  TJ  [x'2+y'*],  for  r,  equal 
to  zero,  is  the  finite  quantity 
2(1  — M).  If  M  is  zero,  x  and  y  are  developable  as  power  series  in  (t  —  £,)1/3, 
and  this  suggests  defining  an  independent  variable  a  in  terms  of  which  x,  y, 
and  t—ti  are  expressible  by  series  of  the  form 

1  (19) 


Fia.  15. 


Since  the  solutions  of  analytic  differential  equations  in  the  vicinity  of 
points  for  which  they  are  regular  are  themselves  regular,  while  in  the  vicinity 
of  singular  points  the  solutions  are  regular  or  not,  depending  on  supple- 
mentary circumstances,  it  is  advantageous  to  choose  such  dependent  vari- 
ables that  the  equations  shall  be  regular  for  rt  =  0,  o-  =  0.  This  Levi-Civita 
has  done,  preserving  the  canonical  form,  with  rare  skill  and  elegance.  His 
dependent  variables,  which  may  be  denoted  here  by  p,  q,  u,  and  v, 
are  related  to  the  rectangular  coordinates  and  their  first  derivatives  by 

*Sur  la  resolution  qualitative  du  probl&ne  restraint  dea  trois  corps,  Ada  Mathematica,  vol.  30  (1906), 
pp.  305-327. 


(  1.0SKI)    oKHll.-    OK    KJKtTIuN    AM)    HKI.A1KI)    PKIUODIC   OHBIIiv 


u— V  —  \v 


I'd 


In  these  variables  equations  (17) 

da  ~       du  dff~        dv 

ff-J 

.  1 


du 
da 


dp 


dv 
da 


_ 
dq 


(21) 


It  follows  from  (20)  that  p  =  q  =  0  if  x+M  =  y  =  0,  and  that  u  and  v  are  finite 
for  r,  =  0.  Therefore  the  form  of  H  shows  that  the  differential  equations  (21) 
are  regular  in  the  vicinity  of  p  =  q  =  0.  Consequently,  p  and  q  are  developable 


Fio.  16. 


i.  17. 


as  power  series  in  a,  vanishing  with  a.  It  follows  from  (20)  that  x,  y,  and 
t  —  (,  ,  considered  as  functions  of  <r,  have  the  form  (19)  ;  and  therefore  x  and  y 
expressed  in  terms  of  T,  defined  in  (6),  have  the  form 

•  •  •  ]•  (22) 


227.  Construction  of  the  Solutions  of  Ejection.—  The  character  of  the 
solutions  in  the  vicinity  of  (<-  0'"=  T  -  °  having  been  found,  they  can  be 
obtained  without  difficulty  from  (17).  Upon  using  r  as  the  independent 
variable  and  expanding  the  expression  for  r,  ,  these  equations  become 


464  PERIODIC    OHBITS. 

It  will  now  be  supposed  that  the  line  of  ejection  is  along  the  z-axis. 
Therefore  the  initial  conditions  are 


-«0,  ~  =  ca  =  arb.  const.       (24) 

or  UT  L         r          JT=O 

With  the  initial  conditions  (24)  the  solutions  have  an  important  prop- 
erty of  symmetry.     Let  them  be  written  in  the  form 


,  =^(T).  (25) 

Now  make  the  transformation  of  variables 


"  l'  "          dr  dn  dr  dr, 

Equations  (23)  are  not  changed  in  form  by  this  substitution.     Therefore 
the  solution  of  the  transformed  equations  with  the  initial  conditions 


r  (0)  -  ij  (0}  -  —  ^l  -  0 
Xl(l    ~yi(l     ~-    U' 


are 


where  /,  y,  <p,  and  ^  are  identical  with  the  functions  represented  by  the 
same  symbols  in  (25).  Therefore 

r2/(r)  =  +r=/(r1)  =  +r/(-r),          r  *>(r)  =  -r^(r,)  =  +r  V(-T),  j 
TY/(T)=-TMrt)=-TM-T),          rV(r)  =  +T^(r1)=-r'V(-r).| 

It  follows  that  x  and  dy/dr  are  even  functions  of  T  and  that  dx/dr  and  i/  are 
odd  functions  of  r.  The  first  equation  of  (22)  contains  only  even  powers  of 
T,  and  the  second  contains  only  odd  powers,  starting  with  a  term  of  the  fifth 
degree  as  the  lowest. 

With  the  initial  conditions  (24),  the  solution  of  (23)  is  found  to  be 


'  (27) 


ty/  |  i,) -i  1/3 
2  >          a  =  arbitrary  constant, 


where  the  positive  or  negative  signs  are  to  be  used  in  the  left  members 
according  as  the  initial  projection  is  in  the  positive  or  negative  direction. 
The  constant  of  the  integral  (18)  is  given  by  the  equation 


(28) 


(U»KH    OHHII.-    n\     K.IK<    IK.N     .\M>    KKI.AIKI)    I'KHK'DH     «.|(|t||>  It,.', 

It  follows  from  (27)  that  the  right  memherof  this  equation  can  be  developed 
as  a  scries  of  the  form 


Since  thi>  equation  must  be  an  identity  in  T  it  follows  that  C  =  C,, 
('-•.=  ('t  =  (\  =  (\=  •  •  •  =0.  Those  expressions  which  are  zero  constitute 
a  check  on  thecompuiationof  t  lie  coefficients  of  (27).  By  direct  substitution 
of  (27)  in  (28),  it  is  found  that 


(29) 
The  force  function  is  sometimes  used  in  the  symmetrical  form 


instead  of  in  the  form  given  in  (17).    Then  the  constant  C  becomes 

(30) 


228.  Recursion  Formulas  for  Solutions.  —  The  second  terms  in  the 
right  members  of  (23)  give  rise  to  a  large  part  of  the  labor  of  constructing 
the  feolutions.  They  can  be  eliminated  by  use  of  the  integral,  and  relatively 
simple  recursion  formulas  can  be  developed  for  the  construction  of  the  solu- 
tion after  the  terms  of  lowest  order  have  been  found. 

The  integral  (18)  becomes  in  the  notation  of  (23) 


(31) 


On  multiplying  the  first  equation  of  (23)  by  Z+M  and  the  second  by  y 
and  adding  the  results;  and  then  multiplying  the  first  by  y  and  the  second 
by  —  (z+/u)  and  adding  the  results,  it  is  found  that 


(32) 


466  PERIODIC    ORBITS. 

On  eliminating  the  second  term  of  the  first  of  these  equations  by  means  of 
the  integral  (31),  the  simplified  equations  become 


=  27Mr6 
The  solution  (27)  may  be  written  in  the  form 


(33) 


[9(1  —  «)H1/'3 
2~^        >         Oj=  —62  =  a  =  arbitrary  constant. 


(34) 


The  next  step  is  to  form  general  expressions  for  the  terms  involved  in  (33). 
Nearly  all  of  the  terms  written  are  of  the  second  degree  in  Z+M  and  y;  those 
which  are  of  degree  higher  than  the  second  are  all  multiplied  by  the  factor 
M  and  will  be  in  general  of  little  importance.  They  will  not  be  included 
in  the  general  formula  and  must  be  added  to  it  when  it  is  used.  Since  these 
terms  contain  TU  as  a  factor,  they  will  not  contribute  much  to  the  solution 
unless  it  is  carried  very  far.  The  general  expressions  for  the  terms  of  the 
second  degree  in  x-\-fi  and  y  are  found  from  (34)  to  be 


;p  =2cv{ 

*- 


l+ 

1  =  1  J=2     *=1 


=  -4cv{l+ 

1  1  =  1  }~2     t=l 

-22/       =  -2cv5r«-2  J  (j+2)6w-6T^+  S  S 

J=4  ;=5     t=l 


{5r«- 

*• 


<  I.(»KI)    incurs    iiK    K.IKI-IIUN    AM)    HKI.AIKI)    I'KHK.Dlr    oHHII.-v         467 


/-»  t-i 


f  (H  -V>M.r+M)'-  -  f  ^l+2M)r'fr'-f-22^.,r-4-  2  J 

1  ;-4  ,-»     »., 


C    l- 

/-T  y-»     *., 


/-I 


.  /-, 


>-»  *-i 


(>+l)(2j+l)a,-6jr" 


-T(x-f  n)%%  =  -2cVJ- 10-  2 
"T  I  TM 

+  2  S  (k+2)(2k+5)alj.ub^, 

n  dx  vT/-  iTiV^vi 

-2y^-  =  -4c1TN-l-  >,    0+1)0^-6^  rw+  X  X1 

Q  7  ^*^^  L  ^^^   ^*^ 


Gr1  yg  =  -6r  r'{5r'-2  2  0>2)6w_,ru+  JJ  2 

*•  ;-4  ;-»   »-i 


2cIr'{-5-  2  [H, 
1  /-i 


^-1     *-! 


468  PERIODIC    ORBITS. 

On  substituting  these  expressions  in  (33)  and  equating  the  coefficients  of 
r*i+3  and  TM+6  respectively,  it  is  found  that 


- 

+62  (2fc+5)6,  A,-*-.+  f  (1  +2M) 


97  ^C^ 

-— (1  —  IJL)  ^bub2J_2k_u-{- quantities  coming  from 
terms  in  (33)  of  the  third  and  higher  degrees,     (35) 


-  12(./+2)62>_6+ 


^-2'-6+  quantities 
*=i 

coming  from  terms  in  (33)  of  the  third  and 
higher  degrees. 

These  formulas  are  to  be  used  when  j  is  3  or  greater,  and  care  must  be  taken 
in  adding  terms  coming  from  the  higher  powers  of  (X+M)  and  y  in  the 
right  members  of  equations  (33). 
The  results  of  applying  (35)  are 


2,418,092    6   «  ,    55,964,945    ,   14  _  38.481,084,886    ,   16  ,  ~| 

3,972,769  '67,540,473  32,937,237,333  J 


(36) 


']' 


CLOSED   ORBITS  OF  EJECTION   AND    HKI..VI  Kl>   PERIODIC  ORBITS.        469 

229.  The  Conditions  for  Existence  of  Closed  Orbits  of  Ejection.  —  The 
series  (34  1  CM  urn-rue  for  all  |M|^/U«  and  |o-o,|£p  provided  \r\£R,  where  R 
is  a  positive  constant  depending  on  ^>,  a,,  and  p.  The  coefficients  of  the 
various  powers  of  T  are  polynomials  in  a  and  M,  so  far  as  M  occurs  explicitly, 
and  they  also  involve  M  implicitly  through  c.  These  coefficients  are  expansible 
as  power  series  in  a  —  a,,  and  n  which  converge  for  all  finite  values  of  |a  —  a,| 
and  for  |//|  <  1.  Therefore  the  expressions  for  Z+M  and  y  are  expansible  as 
power  sn-ie>  in  a-o0  =  /3  and  »,  and  if  |/3|£p,  |M|£P«  the  series  converge  for 
all  \T\  ^R.  They  may  be  written 

X  =  PI(IJ,»;T),      ^=p,03,/i;T),      y  =  p,(0,M;r),      ^=P<(0,M;r),  (37) 

where  /;,,...,  p,  are  power  series  in  /3  and  p. 

Now  o,  will  be  determined  so  that  when  n  is  zero  the  period  from 
ejection  to  collision  shall  be  2  jr.  From  (5)  and  (13)  it  is  found  that  a, 
satisfying  this  condition  is 

.  (38) 


Suppose  T<R  and  /u  =  0  and  that  for  r  =  T  the  coordinates  of  the 
infinitesimal  body  are  xt,  x't,  yol  y't,  where  the  accents  denote  derivatives 
with  respect  to  T.  Suppose  now  0  <ju<  H>  and  let  the  values  of  the  coordinates 
at  r=T  be 

(39) 


The  conditions  that  these  values  of  the  coordinates  shall  belong  to  an  orbit 
of  ejection  are 

T),      y.+A-p.O,  M;  T), 

I    (40) 

,  M;  T). 


Since  the  right  members  of  these  equations  are  expansible  as  converging 
power  series  in  0  and  M,  it  follows  from  the  definitions  of  xt,  z{,  y,,  and  y't  that 

A  =  fc(/3,/0,        A  =  ftO,M),        0,  =  9,G3,  M),        h>~q<(fi,ri,       (41) 

where  9,  ,  .  .  .  ,  <?4  are  power  series  in  0  and  M,  vanishing  with  0  and  M- 

If  the  infinitesimal  body  crosses  the  x-axis  perpendicularly  at  any  time 
the  orbit  is  symmetrical  with  respect  to  the  z-axis.  It  follows  from  the 
definitions  of  xt  ,  x't  ,  y0  ,  and  y'9  that,  when  M  =  0, 


«J(P/2)-0,        y,(P/2)-0,  (42) 

where  P  is  the  period  in  T  from  ejection  to  collision.  It  will  be  shown  that 
analogous  conditions  can  be  satisfied  when  M  is  distinct  from  zero,  and  there- 
fore that  closed  orbits  of  ejection  exist  in  the  restricted  problem  of  three 
bodies. 


470  PERIODIC    ORBITS. 

Suppose  M  is  distinct  from  zero  and  consider  the  solution  with  the 
initial  conditions  x0+^ ,  Zj+Aj ,  2/0+183 ,  2/o+&  •  In  order  to  leave  the  period 
arbitrary  an  undetermined  parameter  5  is  introduced  by  the  transformation 

r  =  rt(l  +  5),  (43) 

where  rl  is  the  new  independent  variable.  Now  if  the  orbit  for  ^  =  0  does 
not  pass  through  the  position  of  fj.  the  solutions  can  be  expanded  as  power 
series  in  /31;  .  .  .  ,  /84,  5,  and  /z;  and  if  Q  is  arbitrarily  chosen  in  advance, 
the  moduli  of  ft ,  .  .  .  ,  j84 ,  d,  and  M  can  be  taken  so  small  that  the  solutions 
converge  for  all  T^r^l  +  ty^Q.  The  quantity  Q  will  be  taken  equal  to 
(l  +  5)P/2  where,  as  before,  P  is  the  period  from  ejection  to  collision  when 
M-0. 

In  order  to  complete  the  discussion  in  regard  to  the  convergence  it  is 
necessary  to  show  that  none  of  the  orbits  in  question  f  or  /*  =  0  passes  through 
the  position  of  M-  Suppose  T  =  TO  is  the  time  at  which  the  infinitesimal  body 
crosses  the  positive  z-axis  on  which,  for  /z  =  0,  the  body  /z  lies  at  the  distance 
unity  from  the  origin.  It  is  necessary  to  show  that  for  none  of  the  values  of 
£0  defined  in  (16)  is  equation  (4)  satisfied  by  £=  1  and  t0  —  tt  =  mr,  where  ^  is 
the  time  of  ejection. 

It  follows  from  (4)  that  the  larger  |£0|  is  the  shorter  is  the  time  required 
for  the  infinitesimal  body  to  pass  from  ejection  to  the  distance  unity. 
Therefore,  if  for  the  smallest  value  of  £0  belonging  to  the  problem,  viz., 
£0=2,  it  reaches  the  distance  unity  in  less  than  TT,  then  it  will  always  be 
at  a  distance  greater  than  unity  at  t—t^ir  and  all  multiples  of  -w  until  it 
reaches  the  greatest  distance  £0 .  Consequently,  it  can  not  pass  through  the 
point  occupied  by  /z  while  receding  from  1  —  /z;  and  since  the  path  referred 
to  rotating  axes  is  symmetrical  with  respect  to  the  z-axis,  it  can  not  pass 
through  the  position  of  /z  on  its  return  to  1— /z.  In  making  the  computa- 
tion it  is  convenient  to  let  £  =  £0P-  Then,  transferring  the  origin  to  tl,  equa- 
tion (4)  becomes 


\3/2 


It  is  found  from  this  equation  that  if  £0  =  2  the  value  of  t  —  ^  for  £=  1  is 
which  is  less  than  TT.     Therefore  none  of  the  orbits  in  question  for  /z  =  0  passes 
through  the  position  of  M- 

If  the  infinitesimal  body  is  moving  in  an  orbit  of  ejection  and  crosses 
the  x-axis  perpendicularly  at  any  time,  then  it  follows  from  the  symmetry 
of  its  motion  that  its  orbit  is  also  an  orbit  of  collision.  Therefore  sufficient 
conditions  for  a  closed  orbit  of  ejection  are 

-i-O  «>,-£• 


If  the  initial  conditions  are  Zo+ft,  x'a+(3t,  y9+03,  t/o+&  and  the  parameter 
5  has  been  introduced  by  (43),  these  equations  become 


CLOSED   ORBITS  OF  EJECTION   AND    I;  D    1'KKIODIC  ORBITS.        471 

^=Fl(0l,...,0t,o,n;P/2)  =  0,    y-Pl(A,...,A,«lM;P/2)-0,    (44) 

where  P,  and  P,  are  converging  power  series  in  0, ,  .  .  .  ,  04,  5,  and  M-  It 
follows  from  (42)  that  P,  and  P,  vanish  for  0,=  •  •  =/34=j=iM«0. 

If  0, ,  .  .  .  ,  04  are  determined  by  (41)  the  orbit  is  an  orbit  of  ejection. 
Therefore,  upon  substituting  the  series  for  these  constants  in  (44),  sufficient 
conditions  for  the  existence  of  closed  orbits  of  ejection  become 

=  Q,(0,  5,  M;  P/2)  =  0,         y  =  Qf(0,  5,  „;  P/2)  =  0,  (45) 

where  Q,  and  Q,  are  power  series  in  0, 5,  and  M,  which  vanish  with  0  =  6  =  n  =  0. 
The  coordinates  can,  therefore,  be  developed  as  power  series  in  0,  3,  and  M 
and  the  moduli  of  these  parameters  can  be  taken  so  small  that  the  series 
converge  for  |T|  <P,  where  P  is  the  period  from  ejection  to  collision  for  /i~0. 

230.  Proof  of  the  Existence  of  Closed  Orbits  of  Ejection. — The  proof 
of  the  existence  of  closed  orbits  of  ejection  resolves  itself  into  the  demon- 
stration that  equations  (45)  have  solutions  when  M  is  distinct  from  zero. 
These  equations  are  not  satisfied  by  n  =  0  unless  0  and  5  are  both  also  zero, 
because,  when  n  =  0,  the  problem  reduces  to  that  of  two  bodies  in  which  the 
period  in  r  from  ejection  to  greatest  distance  depends  upon  0,  and  in  which 
the  distance  depends  upon  5.  Therefore  equations  (45)  have  one  or  more 
solutions  for  0  and  5  as  power  series  in  n,  vanishing  with  n,  according  as  the 
functional  determinant  is  distinct  from  zero  or  is  zero  for  0=  6  =  n**Q. 

Since  the  functional  determinant  involves  derivatives  only  with  respect 
to  0  and  5,  the  /i  may  be  put  equal  to  zero  before  forming  it.  Then  the 

determinant  in  question  is 

dx        dx 
da'       dS 


d£         dy 
da'       dS 


•  -•.-0 

e-t-t 


where  x  is  the  derivative  of  x  with  respect  to  T,  .  Before  forming  the  elements 
of  the  first  column  6  may  be  put  equal  to  zero,  and  before  forming  the 
elements  of  the  second  column  0=a-o,  may  be  put  equal  to  zero. 

It  follows  from  (15)  that  when  6  =/*=  0  and  t—  f,  =  2>T  the  value  of  y  is 
zero  whatever  a  may  be.  Therefore  dy/da  is  zero  and  the  determinant 
becomes  simply  &=(dx/da)(dy/d&). 

In  the  case  under  consideration  the  value  of  t  —  <,  for  which  A  is  formed  is 

«_(1  =  T«  =  TJ(l  +  5)«=-^(l+«)'=;V(l+«)1.  (46) 

The  second  factor  of  A  will  be  computed  first.  Upon  putting  a-o,  =  0  =  0, 
it  is  found  from  the  second  equation  of  (15)  that 


i 
do      L     do  -JJ-» 


472  PERIODIC    ORBITS. 

Before  computing  the  first  factor  of  A  the  parameter  5  may  be  put  equal  to 
zero.     Hence  it  follows,  from  (46),  (15),  and  (9),  that 


It  was  proved  in  §223  that  the  signs  in  this  series  alternate  and  that  a  is 
negative  for  those  orbits  which  lie  entirely  in  the  finite  part  of  the  plane. 
Therefore  dx/da  is  distinct  from  zero  for  all  values  of  a  under  consideration. 

It  follows  from  this  discussion  that  A  is  distinct  from  zero  for 
/3=a-a0=5=0,  T=JT,  and  consequently  that  the  sufficient  conditions  for  the 
existence  of  closed  orbits  of  ejection  can  be  uniquely  satisfied  for  [/*  suffi- 
ciently small.  There  is  a  closed  orbit  of  the  type  in  question  for  ejection 
in  both  the  positive  and  the  negative  direction  for  all  integral  values  of  j 
upon  which  the  £0,  or  a0,  of  (16)  depends. 

In  the  special  case  in  which  the  finite  masses  are  equal,  a  closed  orbit 
of  ejection  for.;'  =  2,  with  ejection  in  the  positive  direction,*  was  discovered 
from  numerical  experiments  by  Burrau  in  two  interesting  memoirs.  f  Since 
in  his  problem  ju  had  the  large  value  0.5,  it  is  not  to  be  expected  that  the 
results  of  this  analysis  would  agree  very  closely  with  the  results  of  his 
computations.  Hence  the  comparison  will  be  made  only  for  the  constant 
of  the  Jacobian  integral.  Upon  taking  into  account  the  difference  in  his 
units  and  those  employed  here,  it  is  found  that  his  Jacobian  constant  CB  , 
equation  (5)  loc.  cit.,  is  related  to  C  of  (29)  by  the  equation 


Burrau's  computation  gave-2CB=  2.2528;  and  for  M=0.5,  ?i?  = 
it  is  found  that  C  =  2.38,  and  the  agreement  is  fully  as  close  as  would  be 
expected.  It  follows  from  these  numbers  that  a  larger  value  of  the  con- 
stant —  a,  corresponding  to  a  smaller  value  of  £„  ,  belongs  to  the  undisturbed 
orbit  having  the  period  2ir  than  to  that  computed  by  Burrau.  In  the  undis- 
turbed orbit  the  greatest  distance  to  which  the  infinitesimal  body  recedes  is, 
by  (16),  £0  =  2;  it  has  this  value  at  t  —  tt  =  w,  and  it  is  then  on  the  negative 
half  of  the  z-axis.  The  greatest  distance  found  by  Burrau  in  his  computation 
was  1.9972,  or  a  little  less  than  that  in  the  undisturbed  motion. 

If  the  infinitesimal  body  is  ejected  toward  or  from  the  body  ^  with  a 
small  value  of  |£0  ,  it  will  be  disturbed  so  that  on  its  return  it  will  revolve 
around  1—  M  in  the  positive  direction.  This  can  be  seen  when  the  motion 
is  considered  in  fixed  axes,  for  under  the  conditions  postulated  the  disturb- 
ance is  positive  all  the  time  that  the  infinitesimal  body  is  going  out  and 
returning.  If  it  is  ejected  farther,  it  will  be  accelerated  by  /z  in  the  negative 

*Burrau's  orbit  of  ejection  was  from  the  body  called  it  here,  but  permuting  1—  p  and  M  and  changing 
the  positive  directions  of  the  axes,  the  statements  are  correct. 

fRecherches  numeriques  concernant  des  solutions  periodiques  d'un  cas  special  du  probleme  des 
trois  corps,  Astronomische  Nachrichten,  vol.  135  (1894),  No.  3230;  and  ibid,  vol.  136  (1894),  No.  3251. 


CI.(»KI)   ORBITS   OF    EJlriK'V    AM)    UKI.  \  I  Kl>    PKKIODIC   ORBITS.         47.'* 

direction  part  of  the  time.  While  in  general  the  body  will  not  collide  with 
1— At  on  its  return,  it  may  possibly  do  so  under  special  conditions.  Indeed, 
Sir  George  Darwin  has  discovered  one  such  orbit  by  numerical  ex|>eriment* 
having  the  period  *-.  The  ejection  was  from  /i  in  the  direction  of  1  —  n,  and 
the  body  collided  with  n  going  in  the  same  direction. t  This  orbit  is  one  of  a 
pair  which  together  are  the  limit  of  certain  periodic  orbit*,  though  they  are 
not  periodic  themselves,  either  pliy>ically  or  mathematically.  The  constant 
('  belonging  to  this  orbit  in  the  units  employed  here  is  20/11  =  1.818.  The 
values  of  the  masses  used  by  Darwin  were  1  —  M=  10/11,  /*=  1/11.  It  follows 
from  (16)  that  {0=2I/>  for  this  period,  and  from  (30)  that  C=  1.716.  In  this 
case  the  £„  belonging  to  the  undisturbed  orbit  is  larger  than  that  belonging  to 
Darwin's  orbit.  The  value  of  £„  is  2'  '=  1.26;  thegreatest  distance  in  Darwin's 
orbit,  according  to  his  diagram,  is  1.3. 

231.  Conditions  at  an  Arbitrary  Point  for  an  Orbit  of  Ejection. — Since 
the  motion  of  the  infinitesimal  body  is  regular  for  all  finite  values  of  r  and 
all  finite  values  of  the  coordinates  except  those  for  which  it  collides  with 
one  of  the  finite  masses,^  it  becomes  a  matter  of  interest  to  determine  in 
any  special  case  whether  the  trajectory  is  one  of  ejection  or  collision  for  a 
finite  value  of  t.  It  is  sufficient,  as  Painleve"  conjectured  and  as  Levi-Civita 
proved, §  that  the  coordinates  and  velocities  shall  satisfy  one  analytic  con- 
dition in  order  that  the  orbit  shall  pass  through  one  of  the  finite  masses  for 
a  finite  value  of  t.  This  conclusion  will  be  established  here  in  a  different  way. 

Suppose  M  is  zero  and  consider  the  problem  of  defining  the  initial  con- 
ditions for  an  orbit  of  ejection  so  that  it  shall  pass  through  the  point  in 
question,  and  so  that  the  components  of  velocity  at  the  point  shall  satisfy  as 
many  conditions  as  possible.  The  velocity  in  rotating  axes  at  any  distance 
from  the  finite  mass  l-n  is  the  resultant  of  the  velocity  with  respect  to 
fixed  axes  and  that  due  to  the  rotation  of  the  axes.  The  velocity  with 
respect  to  fixed  axes  at  any  finite  distance  can  be  made  any  finite  quantity 
by  a  suitable  determination  of  the  constant  a,  or  the  equivalent  constant  & . 
Consequently,  an  arbitrary  speed,  or  one  of  the  components  of  velocity, 
with  respect  to  rotating  axes  at  any  distance  can  be  secured. 

Suppose  the  speed  at  a  given  distance  has  been  assigned;  then  it  is 
possible  to  determine  the  initial  direction  of  ejection  so  that  the  orbit  of 
the  body  will  pass  through  any  point  having  the  given  distance,  for  it  is 
possible  to  do  it  in  fixed  axes  and  the  rotation  simply  changes  the  direction 
of  ejection  by  an  angle  which  is  proportional  to  the  time  required  for  the 
body  to  reach  the  distance  in  question.  It  is  clear  from  this  that  when 
At  =  0  the  conditions  of  ejection  can  be  so  determined  that  the  infinitesimal 

•On  c^UuTf «mUia  of  periodk  orbita,  MonlUg  Notieu  tf  0*  Royal  Agronomical  SM**,  vol.  70 

tS<£  further  remark*  on  thia  orbit  »t  the  do«e  of  |234. 

if'ainlrve,  Lecont  ntr  la  Tktorie  AnalyHque  dtt  Equation*  Dtff,r, nluUrt.  p.  S83. 

\Acta  Mathrmatica,  vol.  30  (1906),  pp.  305-327. 


474  PERIODIC    ORBITS. 

body  shall  pass  through  any  assigned  point  with  any  assigned  speed.  Of 
the  four  quantities  required  to  define  an  orbit,  viz.,  two  coordinates  and  the 
speed  and  direction  of  motion,  three  can  be  taken  arbitrarily  and  the  fourth 
is  determined  by  the  condition  that  the  orbit  shall  be  one  of  ejection.  The 
determination  of  the  fourth  quantity  is  double  because  the  body  has  the 
same  speed  twice,  once  when  it  is  receding  from  1—  M,  and  once  when  it  is 
returning  toward  1—  /x. 

Suppose  that,  for  /x  =  0,  an  orbit  of  ejection  passes  through  the  point 
XT  ,  yT  with  the  speed  VT  =  Vxy+y'r  at  (t  —  t^1^  =  T.  If  1-T  represents  the  speed 
with  respect  to  fixed  axes,  then,  since  the  component  of  velocity  due  to  the 
rotation  of  the  axes  equals  numerically  the  distance  of  the  point  from  the 
origin,  the  relation  between  VT  and  £'T  is 

(47) 


Equation  (2)  determines  £0  ,  the  greatest  distance  to  which  the  body  recedes, 
and  (13)  gives  the  constant  ac.  Equation  (4)  gives  the  value  of  T,  and 
the  direction  of  ejection  is  T  degrees  in  the  negative  direction  from  the  line 
joining  1  —  /x  and  the  point  (XT,  yT}.  Let  the  angle  of  ejection  be  00  . 

Now  suppose  that  /x  is  distinct  from  zero,  but  small.  Let  the  initial 
values  of  a  and  6  be  a0+/3  and  00+7-  Let  a  new  independent  variable  TI 
and  a  parameter  6  be  introduced  by  (43).  Then  the  solution  can  be  written 
in  the  form 

X=PI(P,  7,  5,  M;  O,         y=p3(P,  7,  «,/*;  T,), 


where  Pi  ,  •  •  •  ,  pt  are  power  series  in  /3,  7,  5,  and  /x.    The  moduli  of  these 
parameters  can  be  taken  so  small  that  the  series  converge  for  0  ^  TJ  ^  T. 

The  conditions  that  the  body  shall  pass  through  the  point  (XT,  yT)  with 
the  velocity  VT  at  TI  =  T  are 


l-vT  =  0.  (48) 

Since  these  equations  are  satisfied  by  /3  =  7  =  6  =  ju=0,  they  can  be  written 
as  power  series  in  ft,  7,  8,  and  /x,  vanishing  with  /3  =  7  =  5  =  /x  =  0,  of  the  form 

^03,7,6^;  D  =  0,       Pa(/3,7,6,M;  T)  =  0,       P,(/3,  7,  5,  MJ  T)  =  0.     (49) 

Equations  (49)  are  not  satisfied  by  M  =  0  unless  also  0  =  7  =  5  =  0.  There- 
fore they  have  solutions  for  /3,  7,  and  5  in  terms  of  M  which  vanish  for  ju  =  0. 
If  the  determinant  of  the  linear  terms  in  /3,  7,  and  6  is  distinct  from  zero 
the  solution  is  unique.  In  treating  the  problem  it  is  convenient  to  use 
equations  derived  from  (49)  rather  than  these  equations  themselves.  Let 
(f>  represent  the  angle  between  the  positive  end  of  the  re-axis  and  the  line 
from  the  origin  to  the  point  (xr  ,yT).  Then  let  Q,  ,  Qt,  and  Q3  be  defined  by 

P3.      (50) 


CM  >SKI)    ORBITS   OF   EJECTION    AND    RK1.AIK1)    PKRIODIC   ORBITS.         475 

This  transformation  is  equivalent  to  rotating  the  axes  so  that  (xr,  yr)  lies 
on  the  positive  half  of  the  new  z-axis.     The  solutions  of 


.tf,  7,  «,M;  70=0    (51) 

are  identical  with  those  of  (49),  for  the  two  sets  of  functions  are  linearly 
related  with  non-vanishing  determinant. 

The  determinant  of  the  terms  of  the  Q,  which  are  linear  in  ft,  y,  and  6  is 


(83) 


Before  forming  this  determinant  M  may  be  put  equal  to  zero,  and  before 
computing  the  elements  of  each  column  the  parameters  with  respect  to 
which  the  derivations  are  taken  in  the  other  columns  may  be  put  equal  to 
zero.  When  y=S=n=Q  the  value  of  Q,  is  zero  for  all  values  of  /3;  there- 
fore aQ,/  3/3  =  0.  Since,  for  /i  =  0,  the  distance  of  the  infinitesimal  body 
from  the  origin  is  independent  of  y  it  follows  that  Q,  is  an  even  function  of  y; 
therefore  dQl/dy  =  0  for  0  =  y  =  d  =  n  =  Q.  Also,  since,  for  0  =  5  =  ^  =  0,  the 
velocity  is  independent  of  y,  it  follows  that  dQ^/dy  =  0.  Hence  the  determin- 
ant reduces  to 


aft, 

aft, 

aft 

a/3 

dy 

a« 

aQ, 

aQ,, 

aQ, 

30  ' 

dy 

05 

aQ,, 

aQ, 

aQ, 

a/3 

dy 

d£ 

a& 

~  aT 


aQ, 
3/3 


d/3 


aQ, 

06 
d6 


(58) 


When  M  =  0  the  values  of  x  and  y,  which  are  the  coordinates  referred 
to  rotating  axes,  are 


z  =  £  cos 

where  £  has  the  value  given  in  (9).     Therefore  P,  and  Pt  become 

,  0, 


If  /3  =  5  =  Q,  the  first  terms  of  the  expansions  of  these  expressions  as  power 
series  in  7  are 


Under  the  restrictions  which  have  been  imposed,  <p  =  Ot—T*  and  Q,  becomes 

Q,  =  £(o,+0,  0;  7")7+  '  '  '  '  (64) 

Since  £(a,-fO,  0;  7")  is  distinct  from  zero,  dQt/dy  is  also  distinct  from  zero. 


476  PERIODIC    ORBITS. 

Now  suppose  7  =  M  =  0.     Then  <p  =  d0—T3  and  Ql  becomes 

,  5;  7*)-£(a0+0,  0;  T»).  (55) 


Therefore  dQ1/d{3  =  d£/d(3,  dQl/d5  =  d£/dd,  the  first  of  which  is  positive  by 
the  properties  of  £  which  were  derived  in  §223.  The  second  one  of  these 
partial  derivatives  is  positive  or  negative  according  as,  for  p.  =  j3  =  5  =  0,  the 
infinitesimal  body  is  receding  from,  or  approaching  toward,  the  origin  at 
t  —  ^  =  T3.  If  £  has  its  greatest  value  for  t  —  t^T3,  then  dQ^dd,  which  is  the 
derivative  of  £  with  respect  to  T,  is  zero. 

It  follows  from  (47),  (48),  and  (50)  that,  for  T  =  M  =  0, 


Q3  =  Vl+[r(a0+/3,5;D]2  -  \/l  +  [£'(a0+0,0;T)]2  (56) 

Therefore  the  partial  derivatives  of  Q3  which  appear  in  (52)  are 

a£'(a0+0,5;r) 

dd  _  =        (57) 

,  0;  f)]V 


From  equation  (9)  it  is  found  that 


where  the  signs  of  the  coefficients  alternate.     Therefore  the  partial  deriva- 
tives in  question  are 

'  •  •  ,       (58) 


Since  a0  is  negative,  the  first  of  these  expressions  is  positive;  since  the 
velocity  decreases  or  increases  with  increasing  time  (according  as  the  infini- 
tesimal body  is  receding  from  or  approaching  toward  the  origin)  the  second 
of  these  expressions  is  negative  if  the  body  is  receding  from,  and  positive 
if  it  is  approaching  toward,  the  origin.  If  £'  is  zero  for  t—t^T3,  £  is  an 
even  function  of  5  because  the  motion  is  symmetrical  with  respect  to  T3  as 
initial  time.  In  this  case  the  second  of  (58)  is  zero.  Therefore  if  the 
infinitesimal  body  for  ju  =  /3  =  5  =  0  is  not  at  its  greatest  distance  at  t  —  <t=  T3, 
the  form  of  A  is 

+        +! 
A=    +         I  •  (59) 

where  the  upper  sign  is  to  be  used  if  it  is  receding  from,  and  the  lower  if  it 
is  approaching  toward,  the  origin.  Since  dQ^/dy  is  distinct  from  zero,  A 
is  distinct  from  zero.  Therefore  in  this  case  equations  (48)  are  uniquely 
solvable  for  0,  y,  and  5  as  power  series  in  ju,  vanishing  with  p.  This  means 


'  L08ED   ORBITS   OF   EJECTION    AND    HUM)  I)    PERIODIC  OKHlls.          177 

that  if  an  arbitrary  point  in  the  j-//-plane.  and  a  velocity  greater  than  that 
of  this  point  with  respect  to  fixed  axes,  bo  >electe<l.  then  there  .-\i-t  two 
orbits  of  ejection  passing  through  thi>  point  such  that,  for  »  sufficiently 
small,  the  infinitesimal  body  will  pass  the  point  with  the  given  velocity. 
The  direction  with  which  the  body  passes  the  point  <le]M'nds  upon  the  initial 
conditions,  of  which  it  is  a  regular  analytic  function.  This  is  Painl. 
theorem  for  the  restricted  problem  of  three  bodies. 

If  the  velocity  chosen  equals  that  of  the  arbitrary  point  with  respect 
to  fixed  axes,  so  that,  for  n  =  0,  the  point  in  question  is  at  the  greatest  dis- 
tance to  \\hieh  the  infinitesimal  body  recede-,  then  the  determinant  A  is 
zero  and  the  solution  is  multiple.  The  reason  for  it  is  that  the  two  solu- 
tions which  were  distinct  in  the  other  case  have  united,  and  the  solution 
has  become  double. 

232.  Closed  Orbits  of  Ejection  for  Large  Values  of  p.  —  It  was  proved 
in  §230  that  closed  orbits  of  ejection  exist  provided  |M|  is  sufficiently  small. 
The  question  of  their  existence  for  large  values  of  n  will  now  be  considered. 

If  r  =  r,(l-f5),  the  solutions  of  (23)  may  be  written 

*=/lva,+/3,  «,  MJ  T,),         j/=/,(a,+/3  «,  M;  T,),  (60) 

and  the  conditions  for  a  closed  orbit  of  ejection  with  the  period  2r  in  T,  are 

,M;  *)=o.      (61) 


It  has  been  shown  that,  for  |M|  sufficiently  small,  equations  (61)  can  be  solved 
for  0  and  d  as  converging  power  series  in  M,  vanishing  with  M,  and  that  when 
these  results  are  substituted  in  (60)  the  latter  become  power  series  in  p 
which  converge  for  O^T^T  provided  |M|  is  sufficiently  small. 

Suppose  the  series  for  the  closed  orbit  converge  for  M  =  M,  ,  and  that  the 
values  of  o,+/3  and  d  for  this  value  of  M  are  a,  and  5,  .  The  solutions  of  (23) 
are  expansible  as  power  series  in  T  for  values  of  o,  6,  and  M  in  the  vicinity 
of  a,,  5,,  and  M,.  If  |o-o,|<r,,  |«-*,|<r,  |M-Mil<r,  the  series  converge  if 
T,  <  T,  where  T  is  any  arbitrary  quantity  less  than  the  period  from  ejection 
to  collision.  The  result  will  have  the  form  of  (27)  where  r  is  replaced  by 
T,(l  +  5).  Each  term  of  (27)  can  be  expanded  as  a  power  series  in  a  -a,, 
6  -  5,  ,  M  -  Mi  which  will  converge  provided  |  M  -  Mi  I  <  1  -  M!  •  The  solution 
may  be  written  in  the  form 


X  =  p1(a-ol,  «-3,,/i-Mi;'pi).        l/  =  Pi(a-ai.  5-5,,  M-M,  ;r,),     (62) 

where  p,  and  p,  are  power  series  in  a-a,,  5-6,,  and  M~M,-  Suppose  the 
values  of  x,  y,  and  their  derivatives  at  r^T  for  a-o,  =  5  =  51  =  M-Mi"0 
are  xr  ,  yr  ,  x'r  ,  and  t/'r  .  Let  their  values  for  an  arbitrary  set  of  values  of 
0  _  aj  t  5  _  5t  t  and  M  -  M,  ,  satisfying  the  inequalities  which  insure  convergency  , 


478  PERIODIC    ORBITS. 


be  XT  +  fa  ,  yT  +  ft  ,  x'T  +  ft  ,  and  y'T  +  fa.     Then  ft  ,  .  .  .  ,  ft  are  expansible 
as  power  series  in  a  —  a,  ,  5  —  5t  ,  M  —  Mi  of  the  form 

ft  =  g((a-ai,  5-5lTM-Mi;  T)       (»-!,...,  *),      (63) 

where  the  <?4  vanish  for  a  —  at  =  5  —  5!  =  /x  —  ^  =  0. 

Now  consider  a  solution  with  the  initial  conditions  XT  +  ft  ,  t/r  +  ft  , 
Zr+ft,  2/r+ft-  Suppose  for  ft  =  ft  =  ft  =  ft  =  0  the  infinitesimal  body  does 
not  pass  through  the  position  of  M  for  T^T^TT.  Therefore  the  solutions 
can  be  expanded  as  power  series  in  ft,  .  .  .  ,  ft  which  will  converge  for 
T  5*  T!  (1  +  5)^  ir  provided  the  moduli  of  ft,  .  .  .  ,  ft  are  sufficiently  small. 
At  r1  =  7r  the  expressions  for  the  coordinates  become,  making  use  of  (63), 


X  =  P1(ftl,  .  .    .  ,  ft;  7r)  =  QI(a-al,  6-dlt  M~ 

i/  =  P2(ft,  .  .   .  ,ft;  7r)  =  Q2(a-a!,  5-5t,  M~ 

'  =  P3(ft,  .  .  .  ,ft;  7r)  =  Q3(a-a1,  5-51)M- 

/'  =  P4(ft,  .  .  .  ,  ft;  7r)  =  Q4(o-o1,  d-Slt  M~ 


(64) 


Conditions  that  the  orbit  shall  be  closed  with  the  period  27r/(l  +  5)  are 
Qs(a  —  c^,  5  —  5j,  M— Mi)=0,         Q3(a  —  &1,  5  —  §!,  fj.— Mi)  =  0.         (65) 

It  has  been  seen  that  in  general  the  solution  of  these  equations  for  a  —  a^ 
and  8  —  5j  in  terms  of  ju~  MI>  vanishing  with  M~MI>  is  unique.  This  is  always 
true  unless  the  solution  becomes  multiple.  If  the  multiplicity  is  odd, 
there  is  one  real  solution  for  both  positive  and  negative  values  of  M~ Mi- 
There  are  three  solutions  altogether  for  |ju  sufficiently  small  because  £0, 
defined  in  (16),  has  three  values  for  which  the  conditions  of  a  closed  orbit 
of  ejection  can  be  satisfied,  but  only  one  of  them  is  real.  Consequently 
the  real  solution  can  not  disappear  by  uniting  with  one  of  the  others  unless 
they  first  unite  and  become  real.  Then,  if  two  of  the  real  solutions  should 
unite  and  become  complex,  there  would  be  one  real  one  left.  That  is,  there 
is  one  real  closed  orbit  of  ejection  from  1  — ^  for  all  values  of  /x  from  0  to  1, 
excluding  the  value  unity.  The  argument  has  been  made  for  the  period 
2,ir,  but  it  is  entirely  similar  for  any  multiple  of  2ir. 

It  has  been  tacitly  assumed  in  the  argument  that  none  of  the  orbits 
of  ejection  under  consideration  passes  through  the  position  of  M  for  any 
value  of  fj,;  for  it  was  only  under  this  condition  that  the  convergence  of 
the  series  was  assured.  It  has  been  proved  that  the  closed  orbits  of  ejection 
do  not  pass  through  /x  for  fj.  sufficiently  small.  Since  the  coordinates  in 
these  orbits  are  regular  analytic  functions  of  /z,  it  follows  that  if  any  one  of 
them  passes  through  the  position  of  ^  for  any  value  of  n,  then  for  values 
near  this  one  it  will  pass  near  ^-  The  motion  of  the  infinitesimal  body  in 
the  vicinity  of  a  finite  body  when  referred  to  rotating  axes  is  always  in  the 


CLOSED    ORBITS   OF   EJECTION    AND  RELATKD    PERIODIC   ORBITS.       479 

retrograde  direction,  and  the  orbits  in  question  are  always  symmetrical  with 
respect  to  the  x-axis. 

Consider  the  motion  with  respect  to  M  in  a  closed  orbit  of  ejection  from 
I-M.  Whether  the  ejection  is  toward  or  from  ft  the  motion  with  respect  ton 
in  those  parts  of  the  orbit  which  are  near  to  it  is  direct  instead  of  retrograde. 
Therefore,  the  orbit  can  not  pass  near  ft  without  first  developing  folds  so 
that  a  line  from  n  in  certain  directions  will  intersect  it  three  times.  It 
is  extremely  improbable,  though  not  absolutely  certain,  that  this  sort  of 
development  could  take  place.  It  is  probable  that  the  real  orbits  of  ejection 
which  exist  for  ft  sufficiently  small  continue  in  the  analytic  sense  for  all 
values  of  ft  from  0  to  unity,  and  that  they  do  not  pass  near  ft. 

For  M  =  0  the  orbits  in  which  the  ejection  is  toward  n  are  symmetrically 
opposite  to  those  in  which  the  ejection  is  away  from  ft,  and  these  conditions 


\ 


/ 

\ 

/ 

\ 

4 

\ 

I 

\ 

• 

i 

\ 

\ 

v?2 

i 

>\**5  '•>"•* 

//•/-t 

i 

i 

r 

\ 


Fio.  18. 


KM.    19. 


are  approximately  realized  when  n  is  small.  When  ft  increases,  the  loops 
which  surround  1  —  M  diminish  in  size  and  preserve  their  approximate  forms, 
while  those  which  surround  ft  approach  circles  whose  radii  are  f*,  where 
2jw  is  the  period,  and  whose  centers  are  at  ft.  For  ft  =  1  — «,  where  t  is  very 
small,  the  orbits  have  the  form  shown  in  Figs.  18  and  19.  There  are,  of 
course,  closed  orbits  of  ejection  from  both  of  the  finite  bodies. 

233.  Periodic  Orbits  Related  to  Closed  Orbits  of  Ejection.  There  are 
periodic  orbits  passing  near  one  of  the  finite  bodies  of  which  the  closed 
orbits  of  ejection  are  the  limits.  Suppose  ft  =  0  and  consider  the  motion  of 
the  infinitesimal  body  with  respect  to  the  finite  body  I—/*.  Let  the  infini- 
tesimal body  cross  the  x-axis  perpendicularly  at  t  =  0  near  the  body  1—  it, 
and  let  the  initial  velocity  be  determined  so  that  the  period  is  2r,  or  a  mul- 
tiple of  2ir.  Then  the  motion  with  respect  to  the  rotating  axes  is  periodic, 


480 


PERIODIC    OKBITS. 


the  orbit  is  symmetrical  with  respect  to  the  x-axis,  and  crosses  it  perpen- 
dicularly at  the  half  period.  The  limits  of  these  orbits,  as  the  nearest 
approach  to  1  — M  becomes  zero,  are  the  closed  orbits  of  ejection. 

The  orbits  under  consideration  exist  for  initial  motion  near  the  finite 
body  in  both  the  positive  and  the  retrograde  directions,  but  in  both  cases 
the  motion  in  the  remote  parts  of  the  orbits  when  referred  to  rotating  axes 
is  in  the  retrograde  direction.  Therefore,  those  in  which  the  motion  in  the 
vicinity  of  the  finite  body  is  direct  have  loops,  while  the  others  do  not.  The 
character  of  the  two  classes  of  orbits  for  periods  2ir  and  4?r  are  shown  in  Figs. 
20  and  21.  There  are,  of  course,  orbits  of  a  similar  character  which  are  sym- 
metrically opposite  with  respect  to  the  y-axis. 

Suppose  the  initial  conditions  for  one  of  the  periodic  orbits  in  question 
when  /x  =  0,  are  x(0)  =  o,  z'(0)  =  0,  2/(0)  =  0,  y'(0)  =  b,  and  represent  the 
coordinates  for  this  solution  by  x0,  x'0,  y0,  and  y'a.  The  distance  a  is  small 
and  the  orbit  does  not  pass  through  the  point  (1,  0)  occupied  by  M- 


FIG.  20. 


FIG.  21. 


Now  suppose  n  is  distinct  from  zero  and  that  the  initial  conditions  are 
x(0)  =  a+a,  a;'(0)  =  0,  ?/(0)  =  0,  y'(Q)  =  b+(3.  Let  the  variable  T  be  intro- 
duced by  t  =  T(l  +  d),  where  6  is  an  undetermined  parameter.  Then  the 
solutions  of  the  differential  equations  of  motion,  which  are  regular  functions 
of  the  coordinates  and  5  in  the  vicinity  of  x  =  xa,  x'  =  x'<>,  y  =  y0,  y'  =  y'a,  d  =  0 
for  0  <!  T  ^  j  TT  (j  an  integer),  can  be  written  in  the  form 

,  5,  M;  r),      y  =  y0+i  =y0(r)+pt(a,  0,  d,  /*;  r).| 

(' 


where  pu  •  •  .  ,  p4  are  power  series  in  a,  /3,  and  8,  vanishing  with  a,  0,  and  5. 
Moreover,  the  moduli  of  a,  /3,  and  5  can  be  taken  so  small  that  the  series 
converge  for  0  ^  r  ^jir,  where  j  is  any  integer. 

Sufficient  conditions  that  the  orbit  for  /z  distinct  from  zero  shall  be 
periodic  with  the  period  2jir  are 

pt  (a,  0,  8,  M  J  »  =  0,          pt(a,P,d,n',jir)  =  0.  (67) 


<  l.ii.-KK    i>HHII>    UK    K.IKCIION      \M»    KK.I..VI  K.l>    1'KKIuDH'    uKHI 

The  problem  is  to  show  that  o.  4.  and  6  can  IM-  determined  >o  thai   the-e 
(•((nations  shall   he  satisfied   for  an   arbitrary   n  sufficiently  small.      In   the 
problem  for  ^  =  0,  either  n  or  l>  can  he  taken  arbitrarily  when  the  oth. 
determined  in  terms  of  /  except  for  sign:  one  -ign  belongs  to  the  direct  and 
the  other  to  the  retrograde  orbit.      It  will  be  stip|>os<>d  that  n  is  the  arbitrary. 
Therefore  it  may  be  supposed  that  it  absorbs  the  undetermined  a  and  l< 
only  two  parameters  in  (67)  besides  the  arbitrary  ^. 

Kquations  .117    can  be  solved  uniquely  for  tf  and  5  as  po\\er  -erie*  in  ^, 
vanishing  with  n-  provided 

dp,        .'/'• 


i<  distinct  from  /.en>.  Before  this  determinant  is  formed  n  can  he  put  e<|ual 
to  /em.  and  therefore  A  de|>ends  only  ujxm  the  two-body  problem.  Before 
the  second  column  is  formed  0  can  IM-  put  equal  to  zero.  When  /i  =  #  =  0 
the  period  in  t  is  2jr;  hence  at  the  half  j>eriod  the  infinitesimal  body  is 
on  the  x-axis  and  the  value  of  j  is  an  even  function  of  t-jr.  Now  the 
parameter  5  serves  only  to  vary  the  i>eriod  in  /  (keeping  it  fixed  in  T),  and  i- 
therefore  equivalent  to  varying  t  from  the  half  jx-riod.  When  t  is  nearer, 
«  can  be  determined  so  that  t-jir=jir(l+i)-jr  consistently  with  the 
definition  of  r.  Therefore  ;>,(0,  0,  «,  n\  jv)  is  an  even  function  of  5,  and 
consequently  (")/>,'d5  =  0  for  0  =  5  =  M  =  0-  Hence  the  determinant  6  become- 


The  second  factor  is  distinct  from  zero  because,  except  for  a  constant 
factor,  it  is  the  derivative  of  y  with  respect  to  t  at  r  =jr,  and  this  derivative 
is  distinct  from  zero. 

In  considering  the  first  factor  of  (69)  the  parameter  5  can  be  put  equal 
to  zero.  If  £  and  »>  represent  the  coiirdinates  referred  to  fixed  axes,  the 
expression  for  x  becomes 


Therefore  the  value  of  xf  at  t  =]*  is 


The  problem  is  reduced  to  finding  whether  or  not  the  expression 

^' 

a/3 


r-iv—  -^'+-  (70) 

}  ^ 


is  xero  under  the  a>»umed  conditions. 


482  PERIODIC    ORBITS. 

The  expressions  for  £,  £',  and  ij,  as  given  by  the  two-body  problem,  are 

£  =  a[cosE-e],       ^'=- 


where  E  is  the  eccentric  anomaly,  a  is  the  major  semi-axis  of  the  orbit,  and 
co  is  the  mean  angular  motion  in  the  orbit.  Since  sin£'  =  0  and  cosE=  —  1 
for  t  =jir  and  /3  =  0,  it  follows  that 

d^_       coa   dE         £^_-A| 2^ 

dp~  ~l+e  dp'         dp~^l~e   ~d~p' 

Therefore  (70)  is  not  zero  unless  dE/dp  is  zero.  But  it  is  found  from 
Kepler's  equation  that 

dE      jir    3co 

dp=l-e  ~d~P' 

It  follows  from  the  properties  of  the  two-body  problem  that  at  t  =  0 


From  these  equations  it  is  found  that 


Therefore  neither  factor  of  the  right  member  of  A  is  zero,  and  conse- 
quently the  solution  of  (67)  for  /3  and  8  as  power  series  in  p,  vanishing  with 
M,  is  unique.  When  the  results  of  the  solution  of  (67)  are  substituted  in  (66), 
the  expressions  for  x,  x',  y,  and  y'  become  power  series  in  fj.  alone  (a  having 
been  taken  equal  to  zero)  and  they  are  periodic  with  the  period  2  jir. 

When  /x  =  0  the  limits,  as  a  approaches  zero,  of  the  periodic  orbits 
which  are  being  considered  are  the  closed  orbits  of  ejection.  There  are  two 
families,  depending  upon  the  sign  of  6,  which  have  the  same  limit.  Now 
when  n  is  distinct  from  zero  the  expressions  for  the  coordinates  are  expan- 
sible as  power  series  in  /x,  the  parts  independent  of  ^  are  the  periodic  orbits 
for  n  —  0,  and  the  series  converge,  for  |/z|  sufficiently  small,  while  t  runs 
through  at  least  half  a  period.  Therefore  the  coordinates  for  any  value  of 
t  are  continuous  functions  of  yu-  Since  the  solutions  were  developed  as 
power  series  in  a  they  are  continuous  functions  of  a  for  any  t  and  M  if  a  is 
distinct  from  zero.  But  in  the  variables  of  Levi-Civita  it  is  not  necessary 
to  make  any  restrictions  on  the  initial  conditions.  The  coordinates  are  in 
all  cases  uniformly  continuous  functions  of  the  initial  conditions  for  all  n, 
and  therefore  the  limits  of  the  periodic  orbits  under  discussion  as  a  becomes 
zero  are  the  closed  orbits  of  ejection,  and  this  holds  not  only  for  /i  equal 
to  zero  but  also  for  all  n  sufficiently  small. 


n|{HII>    "|      K.IK.   II..N     \M)    KK1.AIKD    I'KltK  >\>\<     oKHIIS.         483 

2.U.  Periodic  Orbits  having  Many  Near  Approaches.  Tin-  nrl>it>  con- 
sidered in  the  preceding  article  an-  characterized  l>\  the  fart  tlial.  at  least 
for  small  values  of  p,  after  the  infinitesimal  Ixxly  leaves  the  |M>int  neare-t 
1 — n  it  continually  recedes  until  the  mid-|x'ri(xl,  which  is  a  multiple  of  r, 
and  thru  returns  s\  mmetrically.  Orbits  \vill  now  lie  considered  in  which 
the  infinitesimal  body  recedes  fn.ni  and  return-  toward  n  many  times 
Ix-fore  they  re-enter. 

Suppose  /i  is  zero  and  that  the  infinitesimal  body  Ls  started  near  1  — M 
on,  and  ix-rpendicularly  to,  the  line  joining  1— ^  and  n;  and  suppose  the 
initial  conditions  are  Mich  that  its  |x-riod  is  commensurable  with  2*  without 
being  a  multiple  of  '2w.  Let  the  |x-riod  be 

P  =  2i/.  (72) 

where  />  and  q  are  relatively  prime  integers.  Then  the  motion  with  respect 
to  the  rotating  axes  is  periodic  with  the  period 

T  =  Pq  =  2irp.  (73) 

In  thi>  period  the  infinitesimal  Ixxly  runs  through  q  of  its  jxriods  with 
resjx'ct  to  fixexl  axes  and  the  movable  axes  make  p  rotations. 

Now  suppose  that  n  is  distinct  from  zero  and  that  the  initial  conditions 
are  given  slight  variations,  but  of  such  a  character  that  the  infinitesimal 
body  is  started  at  right  angles  to  the  line  joining  the  finite  bodies.  A  new 

indejx'ndent  variable  T  and  a  parameter  5  are 
introduced  by  the  relation  t  —  7(1+6).  Tin- 
solutions  can  be  developed  as  power  series  in 
ft,  6,  and  the  increments  to  the  initial  condi- 
tions, and  the  moduli  of  these  quantities  can 
be  taken  so  >mall  that  these  series  converge 
for  T  £  7V'2.  Then  the  conditions 

*^ 

that  the  solution  shall  be  periodic 
are  that  the  orbit  shall  cross  the 
x-axis  perpendicularly  at  T  =  T/2.  These  condi- 
tions have  the  form  (67)  and  all  of  the  properties 
of  (67)  which  were  used  in  proving  the  existence 
and  character  of  their  solution.  Therefore,  the 
j>eriodic  orbits  which  are  in  question  exist. 

Now  suppose  that  the  initial  distance  from 
1  —  n,  which  was  arbitrary,  is  made  to  approach 

/ero  as  a  limit.  During  this  approach  to  zero  the  distances  to  the  other 
near  apses  vary,  but  there  is  no  apparent  reason  why  all  these  a|>sidal  di-- 
t.-.nces  should  vanish  at  the  same  time.  In  fact,  from  the  lack  of  symmetry 
it  is  doubtful  whether  any  two  of  them  are  simultaneously  zero. 


484  PERIODIC    ORBITS. 

The  simplest  orbits  of  the  type  under  consideration  are  those  for  which 
p  =  l,  <7  =  2.  Their  general  form  for  retrograde  motion  in  the  vicinity  of 
the  finite  body  1  —  /j.  is  shown  in  Fig.  22.  If  the  distance  from  1  — /x  to  a 
becomes  zero,  the  orbit  of  ejection  discovered  by  Darwin  is  obtained.  The 
question  whether  the  distance  from  1  — ju  to  b  becomes  zero  is  one  that  is 
hard  to  answer.  Certainly  it  can  not  be  answered  affirmatively  with  com- 
plete rigor  by  numerical  experiments,  though  the  existence  of  certain  classes 
of  periodic  orbits  can  be  proved  in  this  way.  If,  for  perpendicular  projection 
from  a  given  point  on  the  z-axis  with  a  certain  speed,  the  next  crossing  of 
the  x-axis  is  at  an  angle  which  is  greater  than  ?r/2 ;  and  if,  for  a  perpendicular 
projection  from  the  same  point  with  a  different  speed,  the  next  crossing  is 
at  an  angle  less  than  ?r/2,  then,  from  the  analytic  continuity,  it  can  be 
inferred  that  there  is  an  intermediate  speed  at  which  the  crossing  will  be 
perpendicular.  But  in  the  present  case  these  conditions  are  not  present,  and 
all  that  can  be  said  is  that  when  the  distance  from  1  — ju  to  a  vanishes,  the 
distance  from  1  —p  to  6  is  small,  and  the  approach  to  1  —  n  is  almost  exactly 
along  the  z-axis.  This  is,  of  course,  to  be  expected  from  the  nature  of 
these  orbits. 


CHAPTER  XVI. 

SYNTHESIS  OF  PERIODIC  ORBITS  IN  THE  RESTRICTED 
PROBLEM  OF  THREE  BODIES 

235.  Statement  of  Problem.  In  the  problem  of  two  bodies  then-  arc 
circular  orbits  whose  dimensions  range  from  infinitely  great  to  infinitely 
small.  They  form  a  continuous  scries  geometrically  and  their  coordinates 
are  continuou>  functions  of  the  \an<.ii-  parameters  by  which  they  may  be 
defined.  There  are  orbits  in  which  the  direction  of  motion  is  forward,  and 
others  in  which  it  is  retrograde.  The  two  series  are  identical  only  when  the 
orbits  are  infinitely  great  and  when  they  are  infinitely  small. 

In  the  restricted  problem  of  three  bodies*  the  orbits  which  are  analogous 
to  the  circular  orbits  in  the  problem  of  two  bodies  are  those  which  revol\e 
around  one  or  both  of  the  finite  bodies  and  which  re-enter,  when  referred  to 
rotating  axes,  after  one  >> nodical  revolution.  Those  inclosing  but  a  single 
finite  body  were  treated  in  Chapter  XII,  and  it  was  shown  there  that  the 
delations  from  uniform  circular  motion  are  due  to  the  attraction  of  the 
.•second  finite  mass.  Those  orbits  which  revolve  around  both  finite  masses, 
and  which  are  analogous  to  circular  orbits,  were  treated  in  Chapter  XIII, 
and  it  was  shown  there  that  the  deviations  from  uniform  circular  motion 
are  due  to  the  fact  that  the  finite  masses  are  separated  by  a  finite  distance. 

The  problem  of  the  present  chapter  is  to  trace,  so  far  as  possible,  a  con- 
tinuous series  of  orbits  from  those  inclosing  both  finite  masses  and  having 
infinitely  great  dimensions  to  those  revolving  around  the  two  finite  mosses 
separately  in  orbits  of  infinitesimal  dimensions.  There  are  no  difficulties 
for  very  great  or  for  very  small  orbits,  but  since  in  some  way  the  infinitely 
great  are  eventually  divided  into  two  series  which  become  infinitely  small, 
it  is  clear  that  there  is  a  region  in  which  the  resemblance  to  the  two-body 
problem  is  very  remote.  The  difficulties  arise  in  following  the  orbits  through 
these  critical  forms. 

The  method  of  treatment  is  that  of  analytic  continuation  of  the  solu- 
tions with  respect  to  the  parameters  ujwn  which  they  de|x-nd.  The  differ- 
ential equations  which  define  the  motion  are,  when  referred  to  rotating  a\e< 
and  in  canonical  units, 


(1) 


r,' 


•  The  restricted  problem  of  three  bodies  n  that  in  which  there  arc  two  finite  bodie*  ami  one  infini 
the  finite  bodie*  revolving  in  riroles. 


48<J  PERIODIC    ORBITS. 

These  equations  admit  Jacobi's  integral 


The  general  solutions  of  (1)  can  be  written  in  the  form 

x=f  1(0,1,   .   .   .  ,  a4,  MJ  f),  2/=/2  (ai,  .   .   .   ,  a4,  /x;  t),  (3) 

where  01,  .  .  .  ,  a4  are  the  initial  values  of  x,  y  and  their  first  derivatives. 
The  conditions  that  the  solutions  (3)  shall  be  periodic  eliminate  three  of  the 
four  aj;  and  the  periodic  solutions  have  the  form 

x  =  Ft  (a,  M;  nt),  y  =  F»  (a,  p;  nf),  (4) 

where  a  is  one  of  the  a,,  or  a  function  of  them  (for  example,  the  Jacobian 
constant  C),  and  n  is  a  constant  depending  on  a  and  n  and  so  associated  with 
i  that  the  period  is  2ir/n.  The  problem  under  consideration  is  to  follow 
the  solutions  as  a,  /*,  or  n  varies  through  its  possible  range  of  values.  It  will 
be  found  convenient  in  the  discussion  to  use  sometimes  one  and  sometimes 
another  of  these  parameters  as  independent.  The  correspondence  between 
them  is  not  one-to-one,  so  that  a  series  of  orbits  may  branch  at  a  certain 
point  with  respect  to  one  of  them  and  not  with  respect  to  another.  The 
functions  in  question  are  highly  transcendental  and  it  has  not  been  found 
possible  to  follow  them  with  complete  rigor  through  branch-points  and 
infinities  by  direct  processes.  But  the  fact  that  the  orbits  may  have  a  branch- 
ing with  respect  to  one  parameter  and  not  with  respect  to  another  makes  it 
possible  sometimes  to  establish  the  existence  of  critical  forms  indirectly. 

There  are  orbits  whose  coordinates  are  complex  and  whose  periods  are 
real.  With  varying  values  of  the  parameter  a  they  may  become  real.  This 
does  not  arise  in  the  problem  of  two  bodies.  It  makes  it  necessary  to  include 
in  the  discussion  certain  orbits  which  are  complex  for  the  values  of  the 
parameters  in  terms  of  which  they  are  most  conveniently  expressed. 

Sir  George  Darwin  discovered  several  classes  of  periodic  orbits  by  numer- 
ical experiments.*  Burrau  found  the  limiting  form  of  the  orbits  of  one 
family  of  oscillating  satellites  by  the  same  process.  f  Many  other  families 
have  been  discovered  by  computations  carried  out  in  connection  with  this 
work.  All  these  examples  illustrate  the  theories  to  be  set  forth  here  and 
place  them  on  a  solid  basis  at  points  where  they  fall  short  of  complete 
rigor  because  of  the  difficulty  of  following  orbits  by  analytical  processes 
through  critical  forms. 

Before  taking  up  the  direct  synthesis,  some  of  the  properties  of  the 
periodic  orbits  which  have  been  previously  given  will  be  enumerated  and 
some  additional  ones  will  be  derived. 

236.  Periodic  Satellite  and  Planetary  Orbits.  —  It  was  shown  in  Chapters 
XII  and  XIII  that  there  are  periodic  orbits  encircling  each  of  the  finite  bodies 
separately,  and  others  encircling  both  of  them  together,  which  are  closed 

*  Aetn  Mathematica,  vol.  21  (1897). 

t  Astronomische  Nachrichton,  vol.  135  (1894),  No.  3230;  and  ibid.,  vol.  136  (1894),  Xo.  3251. 


-1M1IIH-    .-I      I'KKKHMt      UUHII-.  1X7 

after  one  synodieal  revolution.  The  direction  of  motion  of  the  finite  bodies 
is  considered  as  iM-ing  forward,  and  tlie  opposite  is  considered  as  l>eing  retro- 
grade. For  small  values  of  the  parameter  in  terms  of  which  the  solutions 
are  develojK-d,  corresponding  to  \t-ry  small  and  very  large  orbit.-  res|>ectively. 
I  here  are  three  classes  in  which  the  motion  i>  direct  and  three  in  which  it 
i>  retrograde.  Only  one  class  of  the  direct  and  one  of  the  retrograde  orbits 
is  real  for  small  values  of  the  parameters.  Darwin's  computations  show 
that  at  least  in  some  cases  the  complex  orbits  may  become  real. 

One  of  the  most  important  properties  of  the  orbits  in  question  is  that 
they  all  cro-s  the  line  joining  the  finite  bodies  perpendicularly  at  every  half 
period.  If  the  parameters  upon  which  the  periodic  orbits  depend  are  varied. 
these  properties  persist,  for  the  solutions  are  expansible  in  integral  or  frac- 
t  ioiial  powers  <  if  the  parameters,  and  the  property  in  question  is  a  consequence 
of  the  character  of  their  coefficients.  Therefore,  the  whole  series  of  orbits 
from  infinite  to  infinitesimal  dimensions  will  possess  this  property,  unle-s 
indeed  they  branch  into  two  series  which  arc  symmetrical  with  resjM-ct  to 
the  line  joining  the  finite  bodies. 

If  the  coordinates  of  a  real  periodic  orbit  are  analytic  in  a  parameter. 
then  the  orbit  can  not  disappear  without  becoming  identical  with  another 
periodic  orbit;*  and  a  complex  orbit  can  not  become  real  without  becoming 
identical  with  another  complex  orbit,  the  two  becoming  real  as  they  become 
identical.  Heal  orbits  disappear  and  appear  in  pairs  with  the  variation  of 
the  parameters  in  terms  of  which  their  coordinates  are  analytic  functions. 

237.  The  Non-Existence  of  Isolated  Periodic  Orbits.—  Appeal  will  be 
made  to  the  numerical  computations  in  establishing  the  existence  of  j>eriodic 
orbits  near  certain  critical  forms.  In  order  that  this  procedure  may  be 
justified,  it  is  necessary  to  prove  that  the  orbits  which  they  have  shown  to 
exist  arc  not  isolated  examples  which  exist  only  for  special  values  of  the 
masses  and  the  other  parameter  on  which  they  depend. 

Suppose  that  for  M  =  MO  equations  (1)  admit  the  periodic  solution 

(5) 


where  the  period  is  2ir/no.    The  initial  conditions  are 

*(0)=*o,       x'(0)=x'0,  y(0)=!/o,       y'(0)=yV          (6) 

The  initial  conditions  determine  the  value  of  the  constant  of  the  Jacobian 
integral 

C,=  -(xV+^+xo'+yo'+^+r  (7) 


The  orbit  in  question  will  not  pass  through  one  of  the  finite  bodies,  for  then 
it  would  not  be  strictly  periodic,  as  was  explained  in  §222.  It  will  not  have 
any  infinite  branches,  for  then  it  could  not  have  a  finite  period. 

•  LM  Mlthodea  Nouvelles  tie  la  M<Vankiue  Cdwte,  vol.  I,  p.  83. 


488  PERIODIC    ORBITS. 

There  are  two  problems  to  be  considered:  (A)  To  determine  whether 
or  not  periodic  orbi ts  exist  for  M  =  MO +X,  C  =  C0,  which  reduce  to  (5)  for  X  =  0; 
and  (B)  to  determine  whether  or  not  periodic  orbits  exist  for  M  =  Mo,  C  =  C0+y, 
which  reduce  to  (5)  for  7  =  0.  If  periodic  orbits  exist  when  the  parameters 
arc  varied  separately,  they  also  exist  when  they  are  varied  simultaneously. 

(A)  Let  M  =  MO+X  and  take  as  initial  conditions 

x'(G)=x'0+a',        y(Q)=y0+ft,        y'(G)=y'0+ft'.     (8) 


Since  the  periodic  orbit  (5)  does  not  pass  through  a  singular  point  of  the 
differential  equations,  the  solutions  can  be  developed  as  series  of  the  form 


(9) 


x  =  Fl  (MO!  WoJ)  +  a  +pi(a,  a',  ft,  ft',  \;  t), 

z'  =  *YGio;  noO  +  a'+p2(a,  a',  ft,  ft',  X;  <), 

y  =  FZ  (Mo;  n0t)+ft  +p3(a,  a',  ft,  ft',  X;  f), 

y'  =  F,'(^;  WflO+0'  +p4(a,  a',  ft,  ft',  X;  t),\ 

where  pi  .  .  .  ,  p4  are  power  series  in  a,  a',  ft,  ft',  and  X  which  ^vanish  identi- 
cally with  these  parameters  and  are  zero  for  t  =  0.     Moreover,  if  any  finite  T 
is  taken  in  advance  the  moduli  of  these  parameters  can  be  taken  so  small 
that  pi,  .  .  .  ,  p4  converge  for  all  0<j^T. 
The  integral  (2)  can  be  written 

F(plt  .   .   .  ,  p4,  a,  a',  ft,  ft',  X)-F(0,   .   .   .   ,  0,  a,  a',  ft,  ft',\)=0,      (10) 

where  F  is  a  power  series  in  pi,  .  .  .  ,  p4,  a,  a',  ft,  ft',  and  X.     Equation  (10)  is 
identically  satisfied  by  pi  =   .  .  .   =  p4  =  0. 

Sufficient  conditions  that  the  solution  (9)  shall  be  periodic  with  the 
period  P  =  2ir/no  are 

p,(a,  a',  ft,  ft',  X;  P)=0,  p8(a,  a.',  ft,  ft',  X;   P)=0,l 

p2(a,  a',  ft,  ft',  X;  P)  =  0,  p4(a,  a',  ft,  ft',  X;  P)=O.J 

It  follows  from  the  form  of  (2)  that  unless  XQ'  is  zero,  equation  (10)  can 
be  solved  for  p-2  as  a  power  series  in  pi,  p3,  p4,  a,  a',  ft,  ft',  and  X,  vanishing 
identically  for  Pi  =  p3  =  p4  =  0.  If  x0'  were  zero  and  y0'  were  not  zero  the 
solution  would  be  made  for  p4  instead  of  for  p2.  If  xa'  and  y0'  were  both  zero 
the  origin  of  tune  would  be  shifted  so  that  at  least  one  of  them  would  be  dis- 
tinct from  zero.  This  is  always  possible  unless  they  are  identically  zero. 
But  they  are  identically  zero  only  when  the  infinitesimal  body  is  at  an  equi- 
librium point,  and  it  has  been  shown  in  Chapters  V  and  IX  that  these 
solutions  are  not  isolated.  Consequently,  it  may  be  supposed  that  (10)  is 
solved  for  p4  in  the  form 

P*,  ?>3j  a,  a',  ft,  ft',  X),  *>((),  0,  0;  a,  a',  ft,  ft',  X)  =  0.      (12) 


Therefore,  if  pi,  p2,  and  p3  are  periodic  with  the  period  P,  then  by  virtue  of 
this  relation  p4  is  also  periodic  with  the  period  P.  Hence  the  fourth  equa- 
tion of  (11)  is  redundant  and  may  be  suppressed. 


M  MIIKsl-    OK    I'KKIoDI.      .,|(lill-  |V.| 

rt  consider  the  solution  of  the  first  three  equations  ,,f  i  1 1 )  for  a,  a', 
and  .i  in  terms  ..f  X.  The  parameter .-/  i-  >ii|ierflui.us  and  m.-.x  be  taken  e«,ll:il 
to  xer...  This  amount-  to  a  definite  determination  of  the  initial  tune.  Sine,- 
i.")j  is  a  real  periodic  solution  t  he  coefficients  of  p,,  p,,  and  p,aren  lUa- 

tions  (11)  are  not  satisfied  by  X  =  0  with  a,  a',  and  ft  arbitrary,  for  then  all 
orbits  would  be  periodic  with  the  same  period  when  M  =  MO-  Similarly,  they 
an-  not  satisfied  by  x  =  ()  unless  all  three  of  the  parameters  a,  a',  and  ft  are 
/ero.  They  are  not  satisfied  by  a  =  a'  =  0  =  0  and  X  arbitrary,  for  then  fixed 
initial  conditions  would  give  a  periodic  orbit  for  all  distribution  of  mass 
between  the  finite  lx)dies.  This  is  certainly  not  true  for  M  "ear /en>.  Tl 
fore,  the  three  equations  can  be  solved  for  o,  a',  and  ,3  as  po\\ 
integral  or  fractional  powers  of  X.  If  the  powers  an-  integral  the  solution 
will  be  unique  and  real  for  both  positive  and  negative  values  of  X.  If  the 
powers  are  odd  fractional,  there  will  be  a  single  real  soli  it  ion  for  both  posit  i  vi- 
and negative  values  of  X.  If  the  powers  are  even  fractional,  there  will  be 
t\\o  real  solutions  for  X  either  positive  or  negative,  de|H-nding  on  the  M^IIS 
of  the  coefficients  of  certain  terms,  and  all  the  solutions  will  Iw  complex  for 
X  negative  or  positive,  depending  upon  the  signs  of  the  same  eoeffici.  m>. 
Hence  in  all  cases  there  are  real  periodic  solutions  for  small  values  of  X 
which  reduce  to  (5)  for  X  =  0. 

(B)  Let  C  =  C0 +y  and  take  the  initial  conditions  (8).     Also  let 

*  =  (l+i)T,  (13) 

where  6  is  an  arbitrary  parameter  and  r  a  new  indejxjndent  variable.     The 
solutions  in  this  case  have  the  form 


(14) 


X  =Fi  (MO;  Wor)  +  a  -t-pi(a,  a',  ft,  ft',  y,  i;  T), 
x  =  FI'(JIQ',  ^o7") ~r"ft'~r"pi(o»  <*',  ft,  ft',  y,  ij  T), 
y  =Ft  (MO;  n«T)+0  +p,(a,  a',  ft,  ft',  y,  S;  T), 
y'  =  Ft'(nt;  n<>T}+ft'+p4(a,  o',  ft,  ft',  y,  i;  T), 

where  p1(  .  .  .  ,  p4  are  power  series  in  a,  o',  ft,  ft',  y,  and  5.  The  parameter  y 
is  determined  in  terms  of  a,  a',  ft,  ft',  and  6  by  equation  (2)  at  r  =  0.  There- 
fore it  may  be  omitted  from  p!(  .  .  .  ,  p4.  Or,  if  y0'  is  not  zero,  equation  (2) 
determines  ft'  as  a  power  series  in  o,  a',  ft,  y,  and  5,  vanishing  with  these 
quantities.  It  will  be  supposed  that  ft'  is  eliminated.  It  follows  from  the 
integral  that  the  last  equation  corresponding  to  (11)  is  redundant,  and  it  will 
be  suppressed.  The  constant  o'  is  superfluous  and  may  be  taken  equal  to 
zero.  Then  the  conditions  that  the  solution  (14)  shall  be  periodic  with  the 
period  P  in  T  are 

p,(o,  ft,  y,  6;  P)  =0,     p,(a,  ft,  y,  5;  P)  =0,     p,(a,  ft,  y,  *;  P)  -0.    (15) 

Consider  the  solution  of  equations  (15)  for  a,  ft,  and  6  in  terms  of  y. 
They  are  not  satisfied  unless  all  four  of  these  parameters  are  zero,  and  con- 
sequently solutions  for  a,  ft,  and  6  as  power  series  in  integral  or  fractional 
powers  of  y  exist,  and  the  circumstances  under  which  they  are  real  are  strict  ly 
analogous  to  those  of  Case  (A). 


490  PERIODIC    ORBITS. 

238.  The  Persistence  of  Double  Orbits  with  Changing  Mass-Ratio  of 
the  Finite  Bodies.  —  In  Chapter  XI,  p.  359,  it  was  stated  that  Darwin's  com- 
putations show  that  the  two  orbits  which  are  complex  for  small  values  of 
m,  or  for  large  values  of  the  Jacobian  constant  C,  unite  and  become  real  for 
a  certain  value  of  C.  In  this  computation  the  ratio  of  the  masses  of  the  finite 
bodies  was  10  to  1.  The  question  naturally  arises  whether  a  corresponding 
double  periodic  orbit  exists  for  other  ratios  of  the  finite  masses. 

The  conditions  for  a  double  periodic  orbit  will  first  be  developed. 
Suppose 

*=/i(0,       *'=//(«),  y=h(C),       y'=f  *'((),         (16) 

is  a  periodic  solution  of  equations  (1)  for  M  =  MO,  having  the  period  P.  All 
orbits  except  those  around  the  equilateral  triangular  points,  which  will  be 
given  special  consideration  in  §242,  are  symmetrical  with  respect  to  the 
x-axis.  In  them  the  origin  of  time  can  be  so  chosen  that  the  initial  con- 
ditions are 

z(0)=*0,  *'(0)=0,  2/(0)=0,  2/'(0)  =  ?/0,         (17) 

and  from  (2),  C  =  C0. 

Now  consider  a  solution  with  the  initial  conditions 


z'(0)=0,      |/(0)=0,      y'(0)=y'o+|8,       C  =  C0+y.     (18) 

The  constant  /8  can  be  expressed  in  terms  of  a  and  7  by  means  of  (2),  and 
it  will  be  supposed  that  j3  is  eliminated  by  this  relation.  Let  a  new  inde- 
pendent variable  T  be  defined  by 

t=(l+d)r,  (19) 

where  5  is  a  parameter  as  yet  undetermined.  Then  the  solution  of  (1)  can 
be  developed  in  the  form 

x  =  pi(a,  7,  5;  T),  z'  =  p2(a,  7,  5;  T), 

y  =  P^(a,  y,  5;  T),  y'  =  p4(a,  7,  8;  T), 

where  pi,  .  .  .  ,  p^  are  power  series  in  a,  7,  and  5. 

The  conditions  that  (20)  shall  be  a  periodic  solution  with  the  period  P 
in  r  are 

,  7,  6;  P/2)=0,  pt(a,  7,  8;  P/2)  =  0.  (21) 


It  will  now  be  shown  that  5  can  be  eliminated  by  means  of  the  second  of  these 
equations.  Suppose  equations  (1)  with  t  as  the  independent  variable  and 
initial  conditions  (17)  are  integrated  as  power  series  in  5.  The  terms  inde- 
pendent of  5  will  be  periodic  with  the  period  P.  Non-periodic  terms  will 
enter  only  when  the  terms  in  the  right  members  at  some  stage  of  the  inte- 
gration contain  terms  with  the  period  P.  Then  t  times  periodic  terms  will 
appear  in  the  solution,  and  terms  multiplied  by  t2  and  higher  powers  of  t 
will  not  enter  until  later  stages  of  the  integration.  Terms  of  this  character 


8YMHI-I- 


\\t\ 


will  actually  M1M,  for  otherwise   tin-  soluti.ui   would   IN-   |N-ri.Nlir   fur  all  *; 

that  is.  the  coordinates  would  In-  constants  \\ith  res|M-ct    to  /.     Tl,. 

sion  for  //  =  /,,  is  an  odd  function  of  I  with  tin-  initial  conditi,,: 

fore  the  (mn  in  /  is  multiplied  by  an  even  function,  that  is,  a  cosine  term 

which  does  not  vanish  at  t  =  P/2.     Therefore  /,,  cam,-  ,,  to  tl,,.  first    Ic.gree 

and  this  parameter  can  Jx-  eliminated,  giving 

**(«,  7)-0,  (22) 

where  /'  is  a  power  >eries  in  a  and  y  vanishing  for  a="y-0. 

SupiH)se  y  is  taken  as  the  independent  parameter  and   that  (2'J     i- 
solved  for  a  in  terms  of  y.     A  necessary  and  sufficient  condition  that     |i, 
I).-  a  double  periodic  solution  with  respect  to  the  Jacol.ian  constant  r  is  that 


-*<•,  T)-0  (23) 

for  a  =  y  =  0.     Suppose  this  condition  is  satisfied. 

Now  suppose  /i  =  /tio  +  X  and  consider  the  (|iie.stion  of  the  existence  of  a 
douMe  periodic  solution  for  this  value  of  ji.  The  double  periodic  solution 
will  exist  provided  the  equation  corresponding  to  (23)  is  satisfied.  Suppose 
the  initial  value  of  x  is 


(24) 

\\h(  re  a  is  &  parameter  which  remains  to  be  determined.  The  steps  cor- 
responding to  those  leading  up  to  (23)  can  be  taken  in  this  case,  and  the 
(•(luatidn  corresponding  to  (23)  becomes 

P,((r,  X)=0  (25) 

for  a  =  y  =  0,  where  Pj  is  a  power  series  in  a  and  X  vanishing  for  <r  =  X  =  0. 
The  series  Pt  contains  terms  in  a  alone;  otherwise,  not  only  would  every 
orbit  whose  initial  conditions  were 

z(0)=*o+<r,          *'(0)=0,          l/(0)=0,          y'(0)=y',,          (26) 

where  z(0)  is  arbitrary,  be  a  periodic  orbit,  but  it  would  be  a  double  periodic 
orbit.  Therefore  (25)  can  be  solved  f  or  a  as  a  power  series  in  X,  or  in  some 
fractional  power  of  X,  vanishing  with  X,  the  multiplicity  of  the  solution  being 
equal  to  the  degree  of  the  term  of  lowest  degree  in  o  alone.  Hence,  in  the 
analytic  sense,  if  there  is  a  double  periodic  orbit  for  X  =  0  (that  is,  for 
M  =  /*O),  then  there  is  a  double  periodic  orbit  for  every  value  of  n  =  ^+\. 
If  the  solution  of  (25)  has  the  form 

<r  =  Xp(X),  (27) 

where  p(X)  is  a  power  series  in  X,  then  there  is  a  single  double  solution  of  the 
series  for  every  )X|  sufficiently  small.  But  if  the  solution  is  in  X',  then  there 
are  two  real  double  solutions  when  X  has  one  sign  and  none  when  it  has  the 
i  >t  her.  That  is,  for  X  =  0  two  real  double  periodic  orbits  unite  and  disappear 


492  PERIODIC    ORBITS. 

by  becoming  complex.  And  in  general,  double  periodic  solutions  appear  or 
disappear  in  pairs,  which  become  identical  for  certain  values  of  fj.,  a  result  analo- 
gous to  Poincar6's  theorem  respecting  the  appearance  and  disappearance  of 
real  periodic  orbits. 

Now  consider  the  real  periodic  satellite  orbits  which  are  re-entrant  after 
a  single  synodical  revolution.  Darwin's  computation  shows  that  for  the 
mass  ratio  of  10  to  1  there  is  one,  and  but  one,  real  double  periodic  orbit 
in  which  the  motion  is  direct.  Hence,  there  is  no  other  double  periodic  orbit 
with  which  it  could  unite  to  disappear  for  any  value  of  M- 

The  same  result  also  is  a  consequence  of  the  analysis  of  Chapter  XI, 
where  it  was  shown  that  for  M  =  0,  and  therefore  for  all  n,  there  are  but  three 
real  and  complex  orbits  of  the  type  in  question.  From  the  fact  that  there 
are  only  three  periodic  orbits  of  the  type  in  question  it  follows  that  there 
can  not  be  more  than  one  direct  double  periodic  orbit;  and  from  its  existence 
for  M  =  Vn>  1— M  =  I%I,  it  follows  that  there  is  one  direct  double  periodic 
orbit  for  all  values  of  n  from  zero  to  unity.  The  orbits  of  inferior  planets 
differ  from  those  of  satellites  only  in  the  ratio  of  the  masses.  Therefore, 
there  are  also  double  periodic  orbits  of  inferior  planets  in  which  the  motion 
is  in  the  forward  direction. 

When  one  of  the  masses  which  revolve  in  circles  becomes  zero,  those 
orbits  around  the  other  which  are  complex  for  small  periods  are  complex  for 
all  periods  from  zero  to  infinity.  In  this  case  there  are  no  double  orbits 
except  those  of  infinite  and  infinitesimal  dimensions.  The  question  arises 
as  to  the  character  of  the  double  orbits  for  very  small  values  of  the  second 
finite  mass.  Consider  the  totality  of  real  circular  orbits  around  one  of  the 
finite  bodies  when  the  mass  of  the  other  one  is  zero.  As  the  second  mass 
becomes  finite,  the  periodic  orbits  are  continuous  deformations  of  the  cir- 
cular orbits  with  the  exception  of  that  circular  orbit  whose  period  is  2ir. 
It  passes  through  the  point  where  the  second  mass  becomes  finite,  and  the 
force  function  for  this  orbit  has  a  discontinuity.  This  means  that  the  orbit 
itself  has  a  discontinuity.  It  is  conjectured  that  the  complex  orbits  have 
corresponding  discontinuities;  that  when  the  second  finite  mass  is  very 
small  there  are  three  real  orbits  about  the  larger  mass  in  which  the  motion 
is  in  the  forward  direction  and  which  have  a  mean  distance  near  unity  and  a 
period  near  2?r;  that  there  are  three  corresponding  real  orbits  of  small  dimen- 
sions about  the  smaller  mass;  and,  finally,  that  there  are  three  similar  orbits 
of  the  nature  of  superior  planets.  For  increasing  values  of  the  Jacobian 
constant  two  of  the  three  in  each  case  unite  and  form  the  double  orbits. 
Consequently,  if  this  conjecture  is  correct,  and  if  the  three  types  of  double 
orbits  are  followed  as  one  of  the  finite  bodies  approaches  zero  as  a  limit,  the 
one  around  the  larger  finite  mass  and  the  one  around  both  finite  masses 
approach  the  unit  circle,  and  the  one  around  the  smaller  finite  mass  approaches 
zero  dimensions. 


M  \  I  UK-MS    c»K     I'KUIMIIH 


The  question   of  direct    double   |M-riodic  orbits   was   |)iit    to   numerical 
te>t  for  n  =  \.     In  all,  ~M  orbits  wen-  computed  for  vari«>u>  values  of  (',  start- 


ing with  C-8.58  and  ending  with  66.      lor 

direct  periodic  orbits  which  were  geometrically 
much  alike,  and  which  intersected  the  ,r-a\i>  near 
.7v~>.  For  smaller  values  of  C  the  correspond- 
ing periodic  orbits  were  more  nearly  identical. 
For  r  =  3.0S«i  they  were  sensibly  identical.  The 

computation  for  this  value  of  C  gave  the  results 
set  forth  in  the  following  table  (Fig.  23): 


;{..r>K  there  wore  two 


*-Hi     C-3.086,      Double  IXTIO.II.  orbit. 

M-M,     C-3XW6,      I)..ul.l.-|«Tf»lic   ..rl.,t 

( 

X 

1 

i' 

.'/' 

J 

z 

V 

*» 

/ 

0 

-7740 

0 

0 

-I   193 

.40 

-5643 

—  .:>. 

m 

-   .317 

(•:, 

-.7693 

-.(. 

IN: 

-1.168 

BO 

-.4893 

-  :«14 

.738 

—    .072 

10 

-  7556 

-.1161 

-1.097 

60 

-.-I17I-. 

-  :ti7n 

aw 

+   .159 

II 

-.7 

-.1684 

I'.IS 

-    .990 

7ii 

-.MIO 

1197 

•H 

M 

.20 

-.7064 

-.•>\\^ 

tat 

-    .862 

Ml 

-.2917 

-.2698 

i.u 

-.6741 

-..' 

ten 

-    .724 

.90 

-.2429 

-  .  1974 

.420 

V.  ; 

.30 

-.6389 

-.2872 

;j-, 

-    .585 

1  00 

-  2097 

-.1048 

1  mfi 

-.6020 

-.31.51 

74B 

-   .448 

1    10 

-.1978 

+  0007 

-.003 

1  DT'i 

There  are  also  three  retrograde  periodic  orbits,  only  one  of  which  is 
real  for  small  values  of  the  parameters  in  terms  of  which  t  heir  coordinates  are 
developed.  The  question  arises  as  to  whether  the  retrograde  complex  peri- 
odic orbits  unite  and  become  real.  In  order  to  test  the  question  by  numeri- 
cal experiment,  20  orbits  were  computed.  For  n  =  \  and  C  =  3.7.r>,  five  orbits 
were  computed,  starting  with  various  values  of  X*  and  determining  j/0'  so 
that  C  should  be  3.75.  The  correspondence  between  x0  and  the  angle  <p 
at  which  the  orbit  crossed  the  z-axis  after  a  half  revolution  is  given  in  the 
accompanying  table: 


M-H, 

C-3.75, 

Motion 

retrograde. 

a 

-.780, 

-.730, 

-.700, 

-.650,     - 

600 

f 

8T, 

85*. 

86°, 

91'30', 

99* 

This  shows,  csix-cially  when  taken  in  conjunction  with 
more  extensive  computations  for  other  values  of  C, 
that  there  is  only  one  retrograde  periodic  orbit  about 
each  of  the  finite  bodies  separately  for  C=3.75. 
Since  only  a  few  retrograde  periodic  orbits  have 
heretofore  been  given,  the  coordinates  for  the  ap- 
proximately periodic  orbit  defined  by  /i  =  J,  C  =  3.75, 
:r0=  —  .650  will  l>e  given  for  enough  values  of  /  to  show  its  geometrical 
characteristics  (Fig.  24). 


I:.      Jl 


494 


PERIODIC    ORBITS. 


M=J^,     C  =  3.75,     Retrograde  periodic  orbit. 

t 

X 

y 

«f 

V 

0 

-.6500 

0 

0 

2.052 

.026 

-.6420 

.0504 

.637 

1.943 

.050 

-.6189 

.0955 

1  .  194 

1.634 

.O7.r> 

-.6835 

.1308 

1.611 

1.173 

.100 

-.5398 

.  1534 

1.854 

.625 

.  125 

-.4922 

.1618 

1.920 

.048 

.150 

-.4451 

.1561 

1.821 

-    .501 

.175 

-.4024 

.1374 

1.582 

-    .982 

.200 

-.3670 

.1078 

1.233 

-1.367 

.225 

-.3414 

.0699 

.807 

-1.637 

.250 

-.3270 

.0266 

.336 

-1.781 

.275 

-.3246 

-.0184 

-    .151 

-1.800 

Eight  retrograde  orbits  were  computed  for  M  — i>  (7=3.214.     In  this 
case  also  the  existence  of  but  one  periodic  orbit  was  indicated,  as  is  shown 


M  =  /4,         C—  3.214,         Motion  retrograde. 

Xe 

-1.0000, 

-    .9000,       -    .8000,       -    .7700,       -    .719, 

-    .685,       -    .650, 

-    .625 

, 

near 
collision 

80°,          83°21',          84°56',        87°41', 

91°15',             96° 

100° 

by  the  accompanying  table  of  correspondences  be- 
tween x0  and  <p. 

The  retrograde  orbit  defined  by  ^  =  \,  (7  =  3.214, 
XQ  =  —  .685,  is  nearly  periodic.  Its  coordinates  for 
various  values  of  t  are  given  in  the  following  table 
(Fig.  25): 


FIG.  25. 


V'-^A,     C-  3.214,     Retrograde  periodic  orbit. 

M  =  H,     C  =  3.214,     Retrograde  periodic  orbit. 

t 

X 

y 

x' 

y' 

t 

X 

y 

x' 

y' 

0 

-.6850 

0 

0 

1.872 

.200 

-.4314 

.1831 

1.684 

-   .577 

.025 

-.6794 

.0461 

.446 

1.815 

.225 

-.3904 

.1640 

1.502 

-    .942 

.050 

-.6630 

.0896 

.867 

1.650 

.250 

-.3559 

.1364 

1.248 

-1.253 

.075 

-.6368 

.1278 

1.221 

1.389 

.275 

-.3285 

.1019 

.936 

-1.497 

.100 

-.6026 

.1586 

1.503 

1.053 

.300 

-.3094 

.0622 

.583 

-1.664 

.125 

-.5624 

.1802 

1.694 

.664 

.325 

-.2995 

.0194 

.207 

-1.747 

.150 

-.5187 

.1916 

1.787 

.247 

.350 

-.2991 

-.0244 

-    .174 

-1.743 

.175 

-.4739 

.1925 

1.782 

-    .174 

Five  retrograde  orbits  were  computed  for  M  =  ?>  (7  =  2.95.  In  this  case 
also  the  existence  of  only  one  periodic  orbit  was  indicated.  The  correspond- 
ence between  x  and  <p  is  given  in  the  following  table: 


M  =  Hi        C  —  2.95,         Motion  retrograde. 

*o 

-1.0000,         -.9000,         -.8000, 

-.719, 

-.625 

V 

81°,            79°10',           82°20', 

89°0', 

103"43' 

M  N  I1IK-I- 


(.UHII-. 


195 


The  (irl)it  defined  l.y  n=\.  ('  =  '>.'*).  j=-.719  is  nearly  |>eriodi<-. 
Its  coordinates  fur  \;iri<m>  value.-  c.f  /  an-  niven  in  tin-  following  table 
(Kin-  --'I'.,  page  -196): 


*-K,    C-2.W 

,-,,rl.,l 

M-.'i,    C-2.95,    RetracrMle  pprioli'-  «.rl.n 

•    Retrograde  |» 

t 

i 

y 

i- 

'< 

t 

X 

V 

*' 

9" 

0 

-.7190 

0 

0 

.718 

.200 

-.6007 

.2146 

1   716 

-    .054 

tat 

-.7140 

IH.M 

.334 

..::, 

at 

-4581 

.2088 

1  i. 

-    .408 

.050 

-.7088 

.0832 

IBB 

.-.77 

m 

-  4171 

I'M.i 

t.ira 

-   .752 

.075 

-.68M 

111  IS 

m 

lir, 

m 

-.3805 

.1715 

1    .Htt 

-    072 

100 

-.'. 

i:..:r. 

1.21'J 

100 

MM 

Illl 

1   140 

-    .:W2 

1  .'.-. 

-.65 

1.427 

.034 

ne 

-.3235 

iota 

- 

in 

•  Ml 

2000 

i  587 

at 

Ml 

IN,  .'7 

IT.", 

-    * 

nie 

1.685 

m 

-.2987 

nisj 

1.-' 

-    Ml 

Two  orbits  \\ere  also  computed  for  C  =  2.75.  The  results  were  similar 
to  those  for  0  =  2.95.  All  the  results  indicate  that  there  is  only  one  real 
retrograde  |>eriodic  orbit  about  each  of  the  finite  Ixxlies. 

239.  Cusps  on  Periodic  Orbits.  —The  orbits  of  ejection  in  a  certain 
>en>e  have  cusps  at  the  point  of  collision  with  a  finite  body.  But  they  have 
been  treated  in  Chapter  XIV  and  require  no  further  comments  here. 

The  coordinates  of  the  infinitesimal  body  can  be  expressed  as  power 
M  •! -ie,s  in  t  —  ti,  if  for  I  =  t\  it  is  at  any  point  for  which  the  differential  equations 
are  regular.  A  necessary  condition  that  the  orbit  shall  have  a  cusp  at  t  =  tt 
is  that  the  expressions  for  both  x  and  y  shall  have  no  linear  terms  in  t  — 1\. 
It  follows  that  if  the  orbit  has  a  cusp  at  *  =  ti,  then  z'(f ,)  =  y'(tt)  =  0.  That 
is,  the  body  is  on  a  curve  of  zero  relative  velocity  at  t  =  ti.  Suppose  its 
coordinates  at  t  =  ti  are  xt  and  y\.  Now  let 

x  =  Xi+£,  y=!/i+ij;  (28) 

then  equations  (1)  become 


M 


496 


PERIODIC    ORBITS. 


The  solution  of  equations  (29)  as  power  series  in  i- 
conditions  £  =  i\  =  £  =  i\'  =  0  is 


24 


B0f.     .  r    A0f.     f  x-,  ,  l 
="^  t~tl)"~~\  t~ti)  " 


ti  with  the  initial 


(30) 


24 


The  direction  cosines  of  the  normal  to  the 
curve  of  zero  relative  velocity  at  the  point  (z.i,  ?/i) 
are  proportional  to  A0  and  B0.  Therefore  the 
tangent  to  the  cusp  is  perpendicular  to  the  curve 
of  zero  relative  velocity  at  the  point  (xi,  y\}. 

Now  take  a  new  set  of  axes  (u,  v  )  with  origin 
at  (xi,  yi),  with  u  having  the  direction  of  the  tan- 
gent and  v  perpendicular  to  it.  If  the  positive  ends 
of  the  new  axes  are  chosen  so  that  the  cosine  and  sine  of  the  angle  from  the 
positive  end  of  the  £-axis  counted  counter-clockwise  to  the  positive  end  of 
the  w-axis  are  proportional  to  A0  and  B0,  and  if  the  positive  end  of  the  u-axis 
is  90°  forward  counter-clockwise  from  the  positive  end  of  the  tt-axis,  then  the 
equations  of  transformation  are 


Fia.  26. 


!-<i)4  + 


,.    |V, 


(31) 


The  value  of  u  is  positive  for  both  positive  and  negative  values  of 
(t  —  ti),  because  the  coefficient  of  (t  —  ti)z  can  vanish  only  at  an  equilibrium 
point.  Therefore  the  positive  end  of  the  w-axis  extends  from  the  point 
(x^  yi)  into  the  region  of  real  velocities.  The  value  of  v  is  positive  for 
small  negative  values  of  t  —  ti,  and  negative  for  small 
positive  values  of  t  —  t\.  Therefore,  if  motion  along  the 
curves  of  zero  relative  velocity  is  taken  as  positive  when  the 
region  of  real  velocity  is  on  the  left,  the  motion  in  the  cusp 
orbits,  in  the  neighborhood  of  the  cusps,  is  in  the  positive 
direction;  that  is,  the  infinitesimal  body  crosses  the  tangent 
to  the  cusp  in  the  positive  direction.  In  Fig.  27,  C  is  a 
curve  of  zero  velocity,  P  is  a  cusp,  0  is  the  orbit  near 
the  cusp,  and  T  is  the  tangent  to  the  orbit  at  the  cusp. 

Now  suppose  the  orbit  having  the  cusp  is  a  periodic  orbit.  If  it  has  no 
other  double  points  than  at  the  cusps,  and  if  it  is  inside  of  the  curves  of  zero 
relative  velocity,  then  it  revolves  around  one  of  the  finite  bodies  in  the 
positive  direction.  If  it  is  outside  of  the  curves  of  zero  relative  velocity,  it 
revolves  with  respect  to  the  rotating  axes  in  the  retrograde  direction.  In- 


FIG.  27. 


M  N  I  1IK-I-    (>K    I'KHluIMc    OKHII-  |'I7 


deed.  all  periodic  orbits  of  superior  planets  revolve  in  tin-  retrograde  direction 
with  respect  to  the  n.tatiiisi  axes.      But  the  ,,rl)it.s  with  cusps,  if  they  have 
DO  Other  double  points.  revolve  in  the  forward  direction  with  ri-|M-rt  t..  • 
ftXee,  because  at   the  cusps  they  have  precisely  tin-  forward  nioli..n  ,,f  the 
rotating;  a  \ 

240.  Periodic  Orbits  Having  Loops  Which  Are  Related  to  Cusps. 

Suppose  the  orbit  defined  by  tin-  initial  conditions 

j(0)  =  .  x'(0)=0,        y(0)=0,        y'(0)-y.',        C-C.,     (32) 

i>  periodic  witli  the  |>eriod  /'.     Therefore 

x'(P/2)=0,  y(P/2)«0. 

Suppose  it  has  a  cusp  at  t  =  ti,  or 

x(tt)=Xl,          *'(«,)  -0         »(«,)  -y,,          y'(<,)«0.  (34) 

If  the  initial  conditions  are  varied  in  such  a  way  that  the  orbit  remains 
periodic,  its  character  in  the  vicinity  of  the  cusp  will  be  changed.  The 
nature  of  these  changes  will  now  he  considered.  Supixwc  the  initial  con- 
ditions are 

x(0)=zo+a,      x'(0)=0,      y(0)=0,      y'(0)=y,'+/3,      C-C,+7;    (35) 
and  that 

<-(l+«)T,  (36) 

\\here  T  is  a  new  inde]M-ndent  variable  and  6  Ls  an  undetcnnined  paramet<T. 
The  solutions  can  be  expanded  as  power  series  in  a,  ft,  and  5.     The 
conditions  that  they  shall  l>e  periodic  are 

z'[(l-H)P/2]  =  p,(a,  ft,  «)-0,         y[(l+*)P/2]  =  p,(«,  ft  3)=0,     (37) 

«  here  /;,  and  p2  are  power  series  in  a,  ft  and  6,  vanishing  identically  with  a. 
0,  and  i.  Now  consider  the  solution  of  (37)  for  0  and  6  in  terms  of  o,  vanish- 
ing with  a.  The  solution  is  always  possible  either  in  integral  or  fractional 
powers  unless  the  equations  are  identically  satisfied  by  /3  =  5  =  0.  But  this 
means  that  all  orbits  in  which  the  infinitesimal  body  crosses  the  x-axis  near 
x0  with  fixed  velocity  y»'  are  periodic,  and  that  they  all  have  the  same 
period.  Since  the  results  arc  analytic  in  o,  the  orbits  may  be  continued  wit  h 
re-|>ect  to  a,  in  the  analytic  sense,  until  the  place  of  crossing,  x,,  is  small. 
But  then  the  methods  of  Chapter  XIV  apply  and  it  is  known  that  the 
period  depends  upon  x0.  Therefore  equations  (37)  are  not  identically  satis- 
fied by  (3  =  5  =  0,  and  they  can  be  solved  for  0  and  6  in  terms  of  o.  The  solu- 
tions will  have  the  form 

^  =  al/'P,(o1/'),  i-a"^.1"), 

where  p  is  unity  if  the  Jacobism  of  PI  and  p*  with  respect  to  ft  and  f>  is  «li>- 
tinct  from  zero  for  o  =  £  =  5  =  0.  If  p  is  not  unity,  it  is  some  other  |*.»itivr 
integer.  In  general  it  will  be  unity. 


498  PERIODIC    ORBITS. 

In  the  computations  of  Darwin  7  was  taken  as  the  parameter  which 
defines  the  orbits.  The  change  can  be  made  here  because  (2)  is  uniquely 
solvable  for  a  as  a  power  series  in  /3,  7,  and  5  unless 


-O  o-          _  n 

*»          ~v»~~  rs      -mu,  2/0  -u. 

'1  ^2 

But  these  equalities  are  satisfied  only  at  one  of  the  collinear  solution  points. 
The  orbits  in  the  vicinity  of  these  points  will  be  omitted  from  this  discus- 
sion because  they  belong  to  quite  another  category. 

If  a  is  eliminated  by  means  of  (2),  and  /8  and  5  by  means  of  (38),  the 
solutions  will  be  expressed  as  power  series  in  71  fp,  and  the  values  of  the  coor- 
dinates at  r  =  ti  are 

^'  x'(r)r=,,  =  0+71/^2,-| 

y'(r)r.tt  =  0+y1/"0J 

where  61,  .  .  .  ,  64  are  power  series  in  71/p. 

The  expressions  for  the  coordinates  in  the  vicinity  of  t  =  tt,  satisfying 
the  relations  (39),  can  be  expanded  as  power  series  in  t  —  1\.  The  solution  is 
found  from  equations  (29)  to  be 


(40) 
•i)    \    •  •  •  ,) 

where 


a2= 

•        -« 


&3  =  -f(l+6)a2+|(l+5)2[JB1a1+JB261],  j 

i 

where  A0,  AI,  A2,  B0,  Blt  B2  are  given  in  (29)  and  P  is  a  power  series  in  7' -'" 
The  transformation  (31)  gives 

u  =  yl/p(02A0  +64Bo)  (r  —  ti) 

(42) 


It  follows  from  (41)  that  for  7  =  0 

M  =  iUo!+5o2)(T-«1)2+  .  .  .  ,         v=  -|(4o2+5o2)(r-^)3+  •  •  •  ,     (43) 

The  equation  v  =  0  determines  the  points  at  which  the  periodic  orbit 
crosses  the  w-axis.  It  follows  from  the  second  of  (43)  that  (t-ti)=Q  is  a 
triple  but  not  a  quadruple  solution  for  7  =  0.  Therefore  there  are  three 
solutions  of  v  =  Q  for  t  —  tt  as  power  series  in  integral  or  fractional  powers  of 
71/p,  vanishing  with  7.  One  of  them  is  simply 

T  -<,  =  ()  (44) 


-1  \  I  HK.-I-    iiK     l'Kl;l«»|)|(      MHHIl-v 


The  other.-.  depend   U|M)II   till'   \allle-  of    the  coejfieielil-   i  .f    the  right 

of  the  >(•<•<,  ii,l  <>iiii:itioii  of  :}'_'..      I'lih-ss  Mo-M*o  =  A"  =  n  for  •)  =  0  the  two 

remaining  solution-.  ha\e  tin-  fr)rm 

-l"+vmK'-'':r       -'—  v/w1"     • 

If  A"  =  ().  which  will  he  exceptionally  if  :it  .-ill.  the  n.rre>|»omling  >..liition- 
<:\i>t  hut  may  he  in  integral  po\\er>  of  7'  '. 

It  has  hem  remarked  that  p  will  in  general  be  unity.  When  it  is  odd 
the  periodic  orhit  with  the  cusp  at  (z,,  yt)  is  a  multiple  orhit.  If  /»  i>  even. 
t  woorhits  which  are  real  when  y  ha>  <>m-  -ign  unite  for  y  =  0  and  disap|x>ar  hy 
becoming  imaginary  when  y  has  the  other  sign.  When  {>  is  odd  then'  i-  a 
-inule  real  orhit  for  7  both  positive  and  negative.  It  is  clear  that  only 
exceptionally,  if  at  all,  will  a  double  periodic  orhit  have  a  cusp.  If  it  v 
so  iii  any  particular  case  the  value  of  ^  could  be  changed,  when  it  would  no 
longer  he  true.  Therefore  it  will  he  sup]>osed  that  f>  is  unity. 

It  follows  from  i4">)  that  the  second  and  third  intersections  of  the  curve 
with  the  w-axis  are  real  for  y  positive  or  negative  according  at<  A'  Is  positive 
or  negative,  and  that  they  are  not  real  when  7  has  the  other  sign.  When 
the  second  and  third  intersections  of  the  curve  are  real  the  curve  consists 
of  a  small  loop  as  is  indicated  in  Fig. 
_'s;  and  when  they  are  complex  the 
curve  has  a  point  near  which  the 
curvature  is  sharp,  as  is  indicated  in 
Fig.  29.  When  there  are  three  inter- 
-ections  of  the  curve  with  the  u-axi-. 
one  occurs  before  t\  and  one  after  /i. 

It  follows  from  this  discussion 

that  if  a  periodic  <>rl>it  for  a  cirtnin  mint'  (\  of  the  Jacobian  constant  has  a 
nis/,.  then  for  n  xliyhlly  larger  (or  smaller)  value  it  has  a  point,  near  a  curve  of 
zero  relative  velocity,  in  the  vicinity  of  tchich  there  is  very  sharp  curmlun;  for 
diminishing  (increasing)  valws  of  ('  the  print  of  xhnrp  cunatnn-  upproachfs 
the  cusp  form  on  the  fv»nv.s-/M»m//m/  ninr  <>/  -rr<>  velocity,  which  it  reaches  for 
C  =  (\;  andfartttill  further  diminishing  (increasing)  values  of  ('  it  ha*  a  small 
loop  near  a  curve  of  zero  velocity.  In  Darwin's  computations  examples  of 
IH-riodic  orbits  with  cusps  were  found.  It  follows,  of  course,  from  the  sym- 
metry of  the  periodic  orbits  with  respect  to  the  ar-axis  that  if  there  is  a  ru*p 
at  x  =  Ji,  y  =  y\,  then  there  is  also  a  cusp  at  z»Zi,  l/=  -y\- 

241.  The  Persistence  of  Cusps  with  Changing  Mass-Ratio  of  the 
Finite  Bodies.  Suppose  for  M  -  Mo  equations  (1)  have  a  periodic  solution 
satisfying  the  initial  conditions 

z(0)  =  x0.  x'(0)=0,  y(0)  =0,  y'(0)-yV         (46) 

and  the  cusp  conditions  at  l  =  l\ 


500  PERIODIC    ORBITS. 

x(t1)=xl!  z'(*i)=0,  y(*0=J/i,  y'(*i)=0.          (47) 

If  P  represents  the  period  of  the  solution,  the  expressions  for  x'  and  y 
satisfy  the  equations 

x'(P/2)=0,  Z/(P/2)=().  (48) 

Now  suppose  M  =  Mo+X  and  consider  the  question  of  the  existence  of  a 
periodic  orbit  having  a  cusp  for  this  value  of  /z-  Let 

t=(l+5)r,        z(0)=z0  +  a,         s'(0)=0,         J/(0)=0,         ?/(0)  =  >/'o+0.     (49) 

The  solution  can  be  expanded  as  power  series  in  a,  0,  5,  and  X.  The  con- 
ditions that  it  shall  be  periodic  in  r  with  the  period  P  are 

x'(r)^,,/2  =  Pl(a,  0,  6,  X)  =0,  7/(r)r=/V2  =  p2(a,  0,  5,  X)  =0,     (50) 

where  pi  and  p2  are  power  series  in  a,  0,  5,  and  X,  vanishing  with  a,  0,  d,  and  X. 
Unless  the  initial  conditions  (46)  define  a  double  periodic  orbit  these  equa- 
tions can  be  solved  uniquely  for  0  and  5  as  power  series  in  a  and  X,  vanishing 
with  a  and  X.  The  results  will  have  the  form 

0  =  9l(a,  X),  «  =  gS(a,X),  (51) 

where  qi  and  q2  are  power  series  in  a  and  X,  vanishing  for  a  =  X  =  0. 

Suppose  0  and  6  are  eliminated  from  the  solutions  by  means  of  equa- 
tions (51).  The  results  will  be  expanded  as  power  series  in  a  and  X.  The 
values  of  the  coordinates  at  r  —  ti  will  be 

l(a,  X),  s'(T)r-«,  =  0+Pi'(a,  X),' 


,  X),  y'(T),-h  =  0+P,'(a,  X) 


where  PI,  PI,  P2,  P2'  are  power  series  in  a  and  X  which  vanish  with  a  and  X. 
The  values  of  the  coordinates  near  r  =  t}  can  be  expanded  as  power 
series  in  r  —  ti  satisfying  equations  (52).     The  results  are 


s'(r)=0  +Pi'(a,  X)+2a(r-<1) 


2/'(r)=0  +P2'(a,  X)+26(T-<0  +   .  .  . 
The  conditions  that  the  orbit  shall  have  a  cusp  at  T  =  t%  are 
0  =  Pi'(o,X)+2a(«a-*1)+  .  .  .  ,  0  =  P2'(o,X)+26(<2-<,)+  ....   (54) 

These  equations  are  not  satisfied  by  X  =  0,  because  a  and  6  contain  terms 
which  depend  upon  xv  and  yi  alone.  Neither  are  they  satisfied  by  a  =  0, 
t2  —  ti  =  Q,  unless  orbits  crossing  the  x-axis  at  x  =  x0  are  periodic  for  all  values 
of  M  and  have  a  cusp  for  the  same  t  —  ti.  But  it  is  known  that  the  points  at 
which  the  periodic  orbits  cross  the  z-axis  depend  upon  the  value  of  M-  There- 
fore equations  (54)  can  be  solved  for  t2  —  ti  and  a  in  integral  or  fractional 
powers  of  X.  In  general  the  solution  will  be  in  integral  powers  of  X.  If  the 


BYMTHMH  <>!•    i-Kitiooir  OKHIIS.  501 

solution  is  in  intr^ntl  cm. 1,1  fractional  powers  of  X,  it  is  real  for  both  |»o>itive 
and  negative  values  of  X.  If  the  solution  is  in  even  fractional  |x>were  of  X, 
there  are  t\\o  real  solutions  \vlu-n  X  has  on.-  sign  and  only  complex  solutions 
when  it  has  the  othrr. 

It  follows  from  this  iliscu>-ion  that  if  a  real  cusp  exists  for  any  value  of 
n,  it  will  exi>t  for  all  other  values  of  ^  unless  two  real  cusps  become  identical 
and  disappear  by  becoming  complex.  Since  an  orbit  N  uniquely  defined 
by  the  condition-  fora  cusp,  as  well  as  by  any  other  initial  conditions,  cusps 
disappear  by  becoming  complex  only  when  two  orbits  become  identical. 

Darwin's  computations  >howed  that  in  the  case  of  one  of  the  orbits 
which  was  complex  for  large  values  of  the  Jacobian  constant  ("satellites of 
( 'lass  C")  there  were  j)eriodic  orbits  without  loops  near  the  cusp  form,  and 
« >\  here  for  smaller  values  of  the  Jacobian  constant  having  loops.  It  follows 
from  the  results  of  §240  that  between  the  two  orbits  there  exists  one  having 
two  cusps  which  are  symmetrically  situated  with  respect  to  the  z-axis;  and 
it  follows  from  the  discussion  of  this  article  that  the  orbits  with  cusps  e\i-t 
for  all  values  of  n  unless  a  cusp  develops  on  another  orbit  which  lat«-r 
becomes  identical  with  this. 

242.  Some  Properties  of  the  Periodic  Oscillating  Satellites  near  the 
Equilateral  Triangular  Points. — In  Chapter  IX,  Dr.  Buck  has  treated  the 
periodic  oscillating  satellites  which  are  near  the  equilateral  triangular  points, 
using  in  a  general  way  the  methods  of  Chapter  V.  It  will  be  necessary  for  t  he 
purposes  of  the  latter  part  of  this  chapter  to  develop  a  few  additional  prop- 
erties of  these  orbits;  and  the  most  important  of  them  can  not  be  established 
by  the  methods  of  Chapter  V,  but  follow  from  the  methods  of  Chapter  VI. 

The  differential  equations  will  be  transformed  by  letting  n  =  f 
For  motion  in  the  vicinity  of  the  equilateral  triangular  points  they  are 


where  X  and  Y  are  of  the  second  and  higher  degrees  in  z,  y,  and  X. 

In  this  article  the  character  of  the  small  oscillations  will  be  discussed. 
In  treating  them  X  and  Y  may  be  provisionally  put  equal  to  zero.  The 
characteristic  equation  on  which  the  nature  of  the  solutions  depends  is 

-^)-0.  (56) 


4'  4^ 

01    9 


The  roots  of  this  equation  are  all  purely  imaginary  or  complex  in  conjugate 
pairs  according  as  l-27/io(l-w)  is  or  is  not  greater  than  /en..  \Vh.-n 
1  -27^(1  -/*>)  is  zero  there  are  two  pairs  of  equal  purely  imaginary  wilut  i«m> 


502  PERIODIC    ORBITS. 

of  (56).  It  will  be  supposed  for  the  present  that  MO  has  such  a  value  that  the 
roots  of  (56)  are  pure  imaginaries  and  distinct,  and  that  M  has  such  a  value 
that  X  =  M  —  MO  is  very  small. 

Let  the  roots  of  (56)  be  +oV^l,  —  <rV~l,  +p\/^T,  —pV^l,  where 
a  and  p  are  real  and  <r<p.  There  are  two  periodic  solutions  of  the  linear 
terms  of  (55),  one  with  the  period  27r/cr  and  the  other  with  the  period  2w  /p. 
If  x  =  Q,  y  =  Ci>Q  at  t  =  0,  the  solution  having  the  period  2ir/<r  is 


x  =     sin  at,  y  =  r-1  sin  at+Ci  cos  at,  (57) 

0i  0i 

where 

b  •      2ff  -3V3  (l-2/io)     n  ,-n 

01-      21     9  >  «1  =       "I—  •>,     <l        <U-  COoJ 

o  ~r  -4-  ff  ~r  4 

The  curve  described  by  the  infinitesimal  body  is  an  ellipse  whose  equation  is 

<*i2+0i2r2    2at      .y2,. 
d2  e?*11*** 

and  if  0  represents  the  angle  between  the  positive  end  of  the  z-axis  and  the 
major  axis  of  the  ellipse,  it  is  easily  found  to  be  defined,  except  as  to  quad- 
rant, by  the  equation 

+l] 

l< 


The  direction  of  motion  in  the  orbit  is  found  to  be  retrograde  from 


There  are  equations  for  the  period  2w/p  which  differ  from  (57),  (58), 
and  (59)  only  in  that  a  is  replaced  by  p,  and  the  subscript  1  on  a,  6,  and 
c  is  replaced  by  2.  The  motion  in  these  orbits  is  also  in  the  retrograde 
direction. 

The  linear  terms  of  equations  (55)  admit  the  integral 


2  =  f  .r2  +(1  -  2Mo)zy  +  ?F  -  <?•  (60) 

Let  the  value  of  C  for  the  orbits  having  the  period  2ir/<r  and  2x/p  be  (?„  and 
Cf.     They  are  found  from  equations  (57)  to  have  the  values 


(61) 

It  follows  from  (56)  that  the  values  of  a-  and  p2  are 


ol-Vl-  27^0(1-  Mo)  .,_l  +  Vl-27Mo(l-Mo) 

~^~  ~'  p  ~2~ 

Since  MO  is  small  <r  is  small  and  the  limit  of  o-2  as  MO  approaches  zero  is  zero. 
On  the  other  hand  p2  is  near  unity,  and  its  limit  as  MO  approaches  zero  is  1. 


B1  N  I  III  -I-    oh     I'KKlnm,      ..Mill- 

Therefore  ('..  is  positive  and  <  i-  negative.  :it  lea-t  f..r  Miiall  values  of 
Mo.  For  the  LftgNOIgiail  equilateral  triangular  solution  xmymO  the  value 
<>f  <'.  =  (',  is  2«vro.  Hence  the  value  of  the  .lacobian  con-taut  I-  greater  for 
the  orl.it-  who>e  period  is  JIT  a.  and  less  for  tho>e  whose  |M-riod  is  '2r  p.  than 
it  is  for  the  Lagrangiaii  e(iuilateral  triangular  |M>int  solution. 

No\\  consider  the  curves  of  zero  relative  velocity.  They  are  known 
to  he  real  only  if  the  value  of  ('  is  greater  than  that  which  belong-  to  the 
equilateral  triangular  point  solution.  Therefore  they  are  real  only  for  the 
solution  with  the  period  '2r  a.  Their  equal  ion  i< 

^'.=i*'+^(i-2*)*j/+!i/'. 

This  is  the  e<  |iiation  of  an  ellipse,  the  direction  of  \\ho-e  major  axis  i>  given  by 

(64) 


The  limit  of  the  right  member  of  this  e\pre>-  ion  for  Mo  =  0  i»  —  v/§. 

Since  a  is  small  when  MO  is  small,  the  approximate  \alue  of  the  risilit 
member  of  equation  (59)  is  —  V3(l  —  2/n).  Therefon-  the  orbit  whose  |>eriod 
i-  JT  a  has  its  axes,  for  small  MO.  nearly  coincident  with  the  axes  of  the 
eorrrespondinK  curves  of  zero  relative  velocity.  The  j-axis  is  in  the  line 
joining  the  finite  body  M  with  the  equilateral  triangular  ix>int,  and  the  other 
i>  of  course  at  right  angles  to  it. 

Ix>t  the  coordinates  in  the  orbit  whose  jieriod  is  2v/a  referred  to  its 
axes  be  $  and  77;  its  equation  is  then 


66) 


A ,  =  I[(a,  cos  0-sin  «)»+&,*  cos1 0], 


c, 


#,  =  ![((!,  sin  0+cos  0)*+&,s  sin1  0]. 
The  corresponding  equations  for  the  curves  of  zero  relative  velocity  are 


cos1  v+~(l  -2Mo)  sin  ^  cos 

in  ^  cos  ^  +  J  cosVl- 


66) 


It  follows  from  (58)  that  when  M«  is  snuill  the  approximate  values 
of  a,  and  &i  are  v/3  3  and  zero  resect  ively.  Sinn-  the  limit  of  9  for 
Mo  =  0  is  -30°,  it  is  found  from  (65)  that 

lira    At      lim    cos*  g+  2V3  sin  g  coe  0+3  sin1  6  _  Q  (6?) 

^  =  0#,=:M«  =  03c<*J0-2v'38in0co80+8int0 


The  limit  of  the  ellipse  for  Mo  =  0  is  a  straight  line  through  the  origin  and 
the  position  of  M.  and  for  small  values  of  Mo  the  eccentricity  i-  near  unity. 


504  PERIODIC    ORBITS. 

The  approximate  value  of  <f  is  also  —30°  when  MO  is  small.  Hence  it  is 
found  from  (66)  that  the  ratio  A/B  also  has  the  same  value  as  Ai/B\. 
Therefore  at  the  limit  the  orbit  with  period  1-w/a  and  the  corresponding 
curve  of  zero  relative  velocity  have  not  only  the  same  orientation,  but  they 
have  also  the  same  eccentricity. 

It  follows  from  (61),  (65),  and  (66)  that  at  the  limit  Mo  =  0 

A.  —  9    ci2  .-  A 
Al      f  C<r 

The  ratio  of  the  dimensions  of  the  periodic  orbit  whose  period  is  2v/tr  to 
that  of  the  curve  of  zero  relative  velocity  corresponding  to  the  same  value 
of  C  is  equal  to  the  square  root  of  this  number,  or  2.  This  is  actually  the 
limit  of  the  ratio  of  the  linear  dimensions  of  the  orbits  to  the  curves  of  zero 
relative  velocity  as  MO  approaches  zero. 

The  discussion  so  far  has  pertained  to  the  linear  terms  alone  of  the 
differential  equations.  The  results  are  the  first  terms  of  the  series  for  the 
periodic  solutions  which  can  be  shown  to  exist  by  the  methods  of  Chapter 
VI.  Consequently,  for  small  values  of  the  parameter  X  they  give  close 
approximations  to  the  periodic  orbits  and  the  corresponding  curves  of  zero 
relative  velocity.  The  period  2ir/<r  is  very  long  for  small  values  of  MO- 

Now  consider  the  periodic  orbits  whose  period  is  ITT/  'p.  The  approxi- 
mate value  of  p  for  small  values  of  MO  is  unity,  and  from  the  equations  corre- 


3  -\ 

spending  to  (58)  it  is  found  that  b2  —  r^,  «•>=  —&%£•     Therefore,  for  these 

lo  lo 

orbits  also 

tan  26  =  -  v/3 

when  MO  has  the  limit  zero.      It  is  found  from  the  equations  corresponding  to 

(65)  that 

Km    At  _  i 
Mo  =  0£2~4' 

Therefore  in  these  orbits  the  length  of  one  axis  is  twice  that  of  the  other. 
The  limit,  for  MO  =  0,  of  the  eccentricity  of  the  orbits  whose  period  is  2w/ff 
is  unity  and  the  limit  of  the  periods  is  infinity;  the  corresponding  limits  for 
the  orbits  whose  period  is  2ir/p  are  \/3/2  and  2ir. 

243.  The  Analytic  Continuity  of  the  Orbits  about  the  Equilateral 
Triangular  Points.  —  The  periodic  solutions  as  developed  by  the  methods  of 
Chapter  VI  are  power  series  in  ^X*,  and  they  involve  MO-  The  coefficients 
of  the  power  series  are  continuous  functions  of  MO-  The  orbits  are  real  when 
X  has  one  sign  and  complex  when  it  has  the  other.  As  X  passes  through  zero 
from  one  sign  to  the  other,  two  real  solutions  for  =*=X-  unite  and  disappear  by 
becoming  complex.  They  do  not  belong  to  the  physical  problem  except 
when  X  =  M—  MO,  but  since  the  orbits  exist  for  every  value  of  MO  distinct  from 
zero  it  is  easy  to  get  an  understanding  of  the  situation  from  the  behavior  of 


M  M  III  -I-    OK    PEUloDK     ..Kill  IS 

the  more  general  .-olutions  when  X  d<xis  nut  equal  /*  —  /•»••     <  h.  -ince  the  • 
ficients  are  continuous  functions  of  ^,.  tin-  parameter  c:in  be  c(>n>idered  as 
varying  \vitli  X  so  that  their  sum  is  n.     That  is.  th<-  solutions  and  the  |>eriod 
can  lie  expressed  in  terms  of  /»  and  X  by  replacing  /io  by  n  —  X. 

The  two  real  orbits  which  unite  and  disappear  for  X  =  0  are  not  geo- 
metrically distinct.  This  appears  to  be  an  exception  to  the  theorem  that 
real  orl>it>  appear  or  disappear  only  in  pairs.  It  arises  because  in  the 
analy.-is  adopted  the  conditions  that  the  orbit  shall  be  periodic  give  a  double 
determination  of  the  same  orbit.  The  two  determination.-,  coincide  when  the 
orbits  shrink  to  zero  dimensions  for  X  =  0.  Such  a  situation  can  arise  only  at 
the  live  equilibrium  points.  Moreover,  the  matter  i.-  quite  different  when 
the  solutions  are  developed  by  the  method  of  Chapter  V.  When  the  para- 
meter t'  passes  through  zero  the  orbits  do  not  disupj>ear.  but  t he  same 861168 
is  obtained  for  both  i>ositive  and  negative  values  of  t',  the  origin  of  time 
belonging  to  a  different  place  on  the  orbit.  In  the  symmetrical  orbits  around 
the  equilibrium  points  which  are  on  the  z-a\i>  the  origin  of  time  is  displaced 
by  half  a  period.  In  the  non-symmetrical  orbit  >  about  the  equilateral 
triangular  ]M>ints  the  origin  is  shifted  from  one  |M>int  where  the  arbitrary 
initial  condition,  e.  g.,  x'(0)=0,  is  satisfied  to  the  other  point  in  the  orbit 
w  hoe  it  is  also  satisfied. 

The  Jacobian  integral  exists  when  the  right  members  of  the  differential 
equations  are  not  limited  to  their  linear  terms.  Hence,  in  place  of  (60), 
the  right  member  is  an  infinite  series  in  x  and  y.  When  the  expressions  for 
x  and  y  as  series  in  X1  are  substituted  the  constant  C  becomes  a  power  serie- 
in  X1,  the  term  of  the  lowest  degree  in  X  being  of  the  first  degree.  Conse- 
quently the  series  can  be  solved  for  X1  as  a  power  series  in  =*C',  and  the 
result  substituted  for  the  solution  in  powers  of  X'  will  give  x  and  y  expressed 
as  power  series  in  <?*,  which  converge  for  |  C  \  sufficiently  small.  AB  C  goes 
through  zero  the  orbits  whose  period  is  2r/<r  change  from  real  to  complex. 
and  those  whose  period  is  2w/p  change  from  complex  to  real.  There  is  a 
branch  on  each  of  the  series  at  C  =  0,  but  the  two  series  are  distinct. 

\\  hen  /m>  satisfies  the  equation 

1-2W1-/*,)=0 

the  values  of  a-  and  p  are  equal  to  V2/2.     In  this  case  four  solutions  branch 
at  X  =  0. 

244.  The  Existence  of  Periodic  Orbits  about  the  Equilateral  Trian- 
gular Points  for  Large  Values  of  it.— The  orbits  heretofore  discussed  have 
been  for  values  of  n  such  that  1  -27n(l  -M)  »s  positive.  If  it  is  zero,  t  here  i> 
a  double  solution  of  zero  dimensions.  Suppose  now  that  n  has  such  a  value 
that  this  function  of  M  is  very  little  less  than  zero,  and  take  *>  so  that 
l-2?Mo(l-A«o)  is  a  little  greater  than  zero.  Then  there  are  real  periodic 
orbits  with  the  periods  2r/<r  and  2r/p  for  X  sufficiently  small.  When  X 


506  PERIODIC    ORBITS. 

is  less  than  /*  —  /*„  these  orbits  do  not  belong  to  the  physical  problem.  The 
analytic  continuation  of  the  solutions  with  respect  to  the  parameter  X  can  be 
made  until  \  =  /J,  —  /JLO  unless  they  have  some  singularity  for  a  real  positive 
value  of  X.  It  is  very  improbable  that  there  is  an  infinity  in  the  solutions 
for  a  real  positive  value  of  X  because  an  infinity  implies  either  an  infinite 
branch  of  the  orbit  or  one  passing  through  one  of  the  finite  bodies.  The 
orbit  could  not  acquire  an  infinite  branch  without  winding  infinitely  many 
times,  in  the  rotating  plane,  about  the  finite  bodies.  Even  if  it  should  pass 
through  one  of  the  finite  bodies  its  continuity  would  be  maintained,  as  was 
seen  in  the  case  of  other  orbits  of  ejection  in  Chapter  XV.  A  branch-point 
would  imply  the  existence  of  other  real  orbits  which  could  become  identical 
with  the  ones  under  consideration.  It  also  seems  very  improbable  that 
there  is  such  a  branch-point  for  small  variations  in  X.  Therefore  it  will  be 
assumed,  as  being  probable,  that  the  periodic  solutions  can  be  continued  to 
the  ones  belonging  to  the  physical  problem  for  X  =  /z  — Mo-  This  applies  both 
to  those  whose  periods  are  Iw/a  and  to  those  whose  periods  are  2ir/p.  The 
possibility  of  their  having  acquired  loops  about  one  or  both  of  the  finite 
bodies  by  having  passed  through  ejectional  forms  must,  however,  be 
admitted.  This  circumstance  makes  numerical  verification  difficult. 

Now  suppose  the  value  of  ju0  approaches  0.0385  ...  as  a  limit,  the  value 
of  fjL0  which  satisfies  1  —  27^(1—  /*<>)  =0,  and  that  the  analytic  continuation 
can  be  made  with  respect  to  X  for  all  MO-  The  expressions  for  the  coordinates 
in  the  two  classes  of  orbits  are  the  same  except  that  a  and  p  are  interchanged. 
As  MO  approaches  0.0385  .  .  .  a  and  p  approach  equality,  and  the  correspond- 
ing orbits  approach  identity  for  X  =  /*  —  /j,0.  A  difficulty  in  attempting  com- 
plete rigor  arises  from  the  fact  that  a  certain  determinant  which  is  distinct 
from  zero  in  the  proof  of  the  existence  of  the  solutions  approaches  zero  as  /x0 
approaches  0.0385  ....  But  if  it  is  admitted  that  the  analytic  continua- 
tion with  respect  to  X  can  be  made  starting  with  any  ^o,  it  follows  that  even 
if  ju  is  a  little  larger  than  0.0385  .  .  .  there  is  a  double  periodic  orbit,  and  it 
surrounds  a  small  real  curve  of  zero  relative  velocity  in  the  vicinity  of  one  of 
the  equilateral  triangular  solution  points.  As  it  is  decreased  toward  the 
limit  0.0385  .  .  .  ,  the  dimensions  of  this  double  periodic  orbit  diminish 
toward  zero  as  a  limit.  There  is  in  this  analysis  a  double  determination  of 
a  double  periodic  orbit,  just  as  of  a  single  periodic  orbit,  and  the  two  deter- 
minations coincide  when  it  has  zero  dimensions.  Consequently  it  can  disap- 
pear at  zero  dimensions  without  uniting  with  another  double  periodic  orbit. 

If  ju  increases  the  double  periodic  orbit  persists,  according  to  the  prin- 
ciples of  §238,  unless  it  becomes  identical  with  another  double  periodic  orbit. 
If  there  were  another  double  periodic  orbit  with  which  it  could  unite  it  would 
envelop  neither  of  the  finite  masses  and  would  have  two  distinct  branches 
which  are  symmetrical  with  respect  to  the  z-axis.  It  is  improbable  in  the 
extreme  that  there  is  another  such  double  periodic  orbit,  which  would  mean 


M  \  1111  -I-    nh     1-1  KIUDIi      ..Kill  t*. 

the  existence  of  four  single  periodic  orhil-  of  tin-  type  under  consideration. 
There  an-  only  two  periodic  orhit>  which  >hrink  mi  tin-  equilateral  triangular 
points  Miul  others  of  the  type  could  arise  only  from  orhits.  which  originally 
had  loo|is  about  a  finite  liody.  pa--in»  thnniuh  :tn  ejectional  form. 

The  existence  of  a  double  periodic  orhit  f(»r  all  values  of  ft  iniplu-  the 
existence  or  two  single  |>eriodic  orhit-  which  liranch  from  it  for  value-  of  the 
parameter-  which  define  the  orhit,  for  example  the  .lacohiaii  con-taut  ('.  il- 
linear  dimen-ion.  or  its  |>eriod.  It  should  In-  added,  of  cour-e.  that  the 
t\vo  serie>  of  orhits  may  branch  at  the  doulile  orhit  when  considered  with 
n-pect  to  one  parameter,  and  form  a  continuoii-  -.-rn--  when  considered 
with  respect  to  another. 

245.  Numerical  Periodic  Orbits  about  Equilateral  Triangular  Points.— 

In  accordance  with  the  principles  of  §244.  two  periodic  orliits  about  (In- 
equilateral triangular  points  should  exist  for  nil  values  of  ^  from  0  to  J, 
and  for  all  values  of  <'  near  that  belonging  to  the  equilateral  triangular 
(•(luililirium  points.  The  only  way  they  could  oe:ise  to  exist  for  at  least  .Minn- 
value  of  ('  would  he  for  all  of  them  to  pass  through  an  ejectional  form  for 
every  ('.  These  orhits  have  an  axis  of  symmetry  only  when  M  =  }-  I'  i- 
very  difficult  to  establish  by  numerical  processes  the  existence  of  a  |>eriodic 
orhit  when  it  has  no  axis  of  symmetry  because,  for  a  given  initial  point, 
there  are  two  arbitrary  comjwmcnts  of  velocity,  and  interpolations  must  he 
made  from  a  two-parameter  family.  Then-fore  the  computations  wen 
restricted  to  the  case  n  =  \.  It  follows  from  the  differential  equations  that 
in  this  case  the  orbits  in  question  have  the  line  z  =  0  as  a  line  of  symmetry. 
Since /x  =  i  is  far  from  the  values  (0^/u^ 0.0385  .  .  .)  for  which  the  existence 
of  the  orbits  in  question  was  established  by  direct  processes,  those  found  by 
computation  can  not  be  expected  to  have  much  geometrical  resemblance  to 
those  found  by  analysis. 

Since  the  surfaces  of  zero  relative  velocity  expand  with  increasing  (' 
and  unite  on  the  jr-axis.  it  follows  that  either  the  |>ericxlic  orbits  about  the 
equilateral  triangular  jxunts  unite  in  pairs  and  disappear  with  increasing 
values  of  C,  or  they  pass  through  the  collinear  equilibrium  points  with 
infinite  periods.  Therefore  it  seemed  best  to  start  computations  for  values 
of  ('  not  much  greater  than  that  belonging  to  the  equilibrium  point,  viz,  3. 

In  attempting  to  discover  j>eriodic  orhits  about  t  he  equilateral  t  riangular 
points  40  orhits  were  computed.  In  17  of  these  C  was  taken  equal  to  :\.(Y.\: 
in  10  it  was  taken  equal  to  3.20;  in  the  remaining  7  it  was  taken  equal  to 
3.284.  The  initial  values  of  the  coordinates  and  eom|>onent«  of  velocity 
were  x«  =  0,  T/«  arbitrarily  chosen,  j/0'  =  0,  x0'  determined  so  as  to  give  tin- 
adopted  value  of  C.  The  computation  was  continued  until  x  became  again 
equal  to  zero,  and  the  approach  to  periodicity  was  determined  by  the  approxi- 
mation of  y0'  to  zero. 


508 


PERIODIC    ORBITS. 


C  =  3.03,  Period  =2X4.388  =  8.776  (Fig.  30). 

I 

X 

y 

x' 

V 

t 

X 

y 

x' 

y' 

0.0 

0 

.1200 

-1.060 

0 

1.2 

-1.0287 

-  .5841 

-  .708 

-  .070 

0.05 

-  .0533 

.1216 

-1.076 

.067 

1.4 

-1.1768 

-  .5687 

-  .762 

+  .222 

0.10 

-  .1083 

.1266 

-1.127 

.131 

1.6 

-1.3285 

-  .4967 

-  .745 

.495 

0.15 

-  .1667 

.1345 

-1.219 

.185 

1.8 

-  .4701 

-  .3726 

-  .660 

.740 

0.20 

-  .2311 

.1446 

-1.366 

.214 

2.0 

-  .5890 

-  .2032 

-  .521 

.947 

0.25 

-  .3047 

.  1549 

-1.593 

.182 

2.2 

-  .6755 

+  .0030 

-  .338 

1.107 

0.30 

-  .3921 

.1600 

-1.919 

-  .022 

2.4 

-  .7221 

.2361 

-  .124 

1.216 

0.35 

-  .4966 

.  1454 

-2.217 

-  .661 

2.6 

-  .7238 

.4856 

+  .108 

1.271 

0.40 

-  .6014 

.0865 

-1.798 

-1.661 

2.8 

-  .6786 

.7409 

.344 

1.274 

0.45 

-  .6685 

-  .0060 

-  .927 

-1.897 

3.0 

-  .5865 

.9917 

.575 

1.226 

0.50 

-  .7019 

-  .0957 

-  .479 

-1.674 

3.2 

-  .4497 

1.2285 

.790 

1.134 

0.6 

-  .7353 

-  .2397 

-  .273 

-1.233 

3.4 

-  .2720 

1.4429 

.982 

1.003 

0.7 

-  .7638 

-  .3475 

-  .313 

-  .941 

3.6 

-  .0586 

1.6274 

1.146 

.838 

0.8 

-  .7995 

-  .4305 

-  .403 

-  .725 

3.8 

-  .8158 

1.7762 

1.277 

.646 

0.9 

-  .8445 

-  .4937 

-  .498 

-  .543 

4.0 

-  .5503 

1.8846 

1.371 

.  435 

1.0 

-  .8986 

-  .5396 

-  .583 

-  .378 

4.2 

-  .2699 

1.9495 

1.427 

.212 

1.1 

-  .9605 

-  .5695 

-  .654 

-  .221 

4.4 

+  .0177 

1.9689 

1.443 

-  .018 

C=3.03,  Period  =2X5.95  =  11.90  (Fig.  31). 

I 

X 

V 

of 

y' 

t 

X 

y 

x' 

y' 

0 

0 

.7428 

.074 

0 

2.3 

-  .7940 

-  .5196 

-  .323 

-  .605 

.1 

.0072 

.7406 

.070 

-  .044 

2.4 

-  .8320 

-  .5766 

-  .434 

-  .500 

.2 

.0137 

.7340 

.058 

-  .087 

2.5 

-  .8804 

-  .6198 

-  .532 

-  .365 

.3 

.0185 

.7232 

.038 

-  .130 

2.6 

-  .9380 

-  .6496 

-  .616 

-  .231 

.4 

.0210 

.7082 

.010 

-  .170 

2.7 

-1.0030 

-  .6660 

-  .684 

-  .097 

.5 

.0202 

.6892 

-  .026 

-  .209 

2.8 

-1.0742 

-  .6689 

-  .735 

+  .039 

.6 

.0154 

.6664 

-  .070 

-  .245 

3.0 

-1.2272 

-  .6340 

-  .784 

.312 

.7 

.0060 

.6402 

-  .121 

-  .278 

3.2 

-1.3828 

-  .5459 

-  .762 

.570 

.8 

-  .0090 

.6109 

-  .180 

-  .308 

3.4 

-1.5275 

-  .4076 

-  .675 

.808 

.9 

-  .0303 

.5788 

-  .247 

-  .332 

3.6 

-1.6490 

-  .2249 

-  .532 

1.018 

1.0 

-  .0588 

.5446 

-  .323 

-  .352 

3.8 

-1.7222 

-  .0054 

-  .343 

1.174 

1.1 

-  .0954 

.5086 

-  .411 

-  .367 

4.0 

-1.7841 

+  .2412 

-  .121 

1.284 

1.2 

-  .1414 

.4713 

-  .514 

-  .378 

4.2 

-1.8095 

.  5047 

+  .120 

1.342 

1.3 

-  .1988 

.4329 

-  .638 

-  .390 

4.4 

-1.7358 

.7742 

.368 

1.344 

1.4 

-  .2702 

.3925 

-  .793 

-  .416 

4.6 

-1.6376 

1.0390 

.612 

1.296 

1.5 

-  .3587 

.3482 

-  .986 

-  .489 

4.8 

-1.4920 

1.2892 

.842 

1.199 

1.6 

-  .4683 

.2906 

-1.204 

-  .696 

5.0 

-1.3024 

1.5157 

.049 

1.060 

1.7 

-  .5951 

.1988 

-1.269 

-1.199 

5.2 

-1.0744 

1.7107 

.227 

.884 

1.8 

-  .6994 

.0492 

-  .710 

-1.700 

5.4 

-  .8140 

1.8674 

.371 

.679 

1.9 

-  .7384 

-  .1158 

-  .157 

-1.522 

5.6 

-  .5288 

1.9806 

.475 

.450 

2.0 

-  .7459 

-  .2519 

-  .042 

-1.209 

5.8 

-  .2268 

2.0464 

.538 

.205 

2.1 

-  .7522 

-  .3604 

-  .100 

-  .972 

6.0 

+  .0834 

2.0623 

.557 

-  .024 

2.2 

-  .7674 

-  .4483 

-  .207 

-  .792 

Fia.  30. 


Fio.  31. 


HK 


sog 


It  \v:is  found  that  for  r --;{.(>:{  there  are  t\\n  jMTiodic  <»rl.it.s  al>«.iit 
the  equilateral  triangular  points  din"erin»;  n>n-ideral>ly  in  dimensions  and 
IM-riod.-:  for  ('  :\:20  there  an-  al-<>  t\v<>  periodic  orbits  which  differ 


C-3.20,  IVriod  -2X4.68-9.36  (Fi«.  8»). 

t 

i 

> 

f 

* 

t 

i 

v 

•r 

1 

n 

n 

.2030 

-   .883 

0 

1.6 

-    2334 

—    .4760 

-   .683 

H 

-     0449 

aou 

-   .904 

1    x 

—   .8008 

—    .3976 

-   .831 

Ml 

Hi 

-    .0908 

10M 

-    .936 

OM 

-     4772 

-    .2780 

-   .537 

U 

—      i:i'«i 

-Mill 

- 

001 

-     '. 

-    .1146 

-   .408 

-ii 

-      1"«I7 

2152 

-     07., 

.104 

-.'  1 

_     ,. 

+     ' 

-   .346 

014 

-    .'.M7J 

-      I'M 

.-,x., 

INK) 

;M) 

-    .3KM 

- 

—     661'J 

+ 

i:tx 

-    .4l.lx 

2110 

- 

- 

-    .6136 

Ml 

-      • 

-      17  1 

-1 

- 

H,   ,.' 

n 

m 

-    .71150 

-    .01  IJ 

-      • 

-1.470 

34 

_      i. 

1741 

Ml 

7 

-    .1484 

-    :t7.' 

-1  199 

-     2458 

Ml 

004 

1 

-   !8181 

-    .2548 

-    .320 

-    .940 

-     0588 

H4I 

(•M 

-     .KM', 

- 

- 

-    ,7:tl 

40 

-    .8470 

6736 

111 

010 

1    II 

-    .8901 

—     4022 

-       llx 

- 

1  J 

-     6154 

-Kin 

l"x 

1    1 

- 

-    !4498 

-    .484 

-    .399 

4.4 

-     3696 

S537 

i.  a 

-    .9864 

-      1 

- 

-    .254 

46 

-    .1150 

MM 

.284 

OHO 

i  i 

-1.1036 

-   .6065 

-    .626 

+  .018 

4.8 

+     1426 

Moa 

Ml 

-   .098 

C-3.20,  Period-2X6.72-11.44  (Fig.  33). 

1 

z 

V 

z' 

f 

t 

z 

• 

z' 

f 

o 

0 

MM 

-    .188 

0 

22 

-     0165 

-     5708 

- 

-   .129 

M 

-    .0003 

.5549 

-    .1x7 

-   .020 

2.4 

-    .1359 

-     ' 

- 

+  .114 

10 

:,-,;( 

-    .I'M 

-     040 

26 

-     2650 

-   .6266 

- 

.15 

-    !0285 

MOO 

-    .199 

-    .060 

2.8 

- 

-      1 

-    .Ml 

.569 

.20 

-   .0387 

.5474 

-    .209 

-    .080 

:j  o 

-    .6059 

-    .3997 

-    .520 

.763 

.25 

BOO 

—     fft 

-    .098 

3.2 

-     5970 

-    .1307 

- 

.30 

-     0609 

-    .238 

-     117 

3.4 

-    .6577 

+    0660 

-    .218 

OM 

I 

-    .0867 

6341 

-    .2NO 

-     l.v.' 

3.6 

-    .6827 

.2S1S 

—    .029 

111 

-   .117:! 

1073 

- 

-    .184 

38 

-    .6690 

107V 

+   .168 

i  ;•> 

-    .1542 

.4874 

-    .406 

-   .214 

4.0 

-   .6168 

....; 

u.; 

.7 

-   .1990 

.4645 

-    .494 

-     244 

42 

-    .6243 

•on 

.549 

OM 

.8 

-   .2537 

IHS 

-    .603 

-    .278 

44 

-    .3974 

1600 

718 

Ml 

9 

-     3205 

11  IX-1 

-   .737 

-    .331 

4.6 

-    .2388 

ua 

•M 

Ml 

1.0 

-    .4019 
-   .4989 

1708 

.3196 

-   .893 

-1.050 

-    .428 
-    .618 

4.8 
5.0 

-    .0531 
-   .8453 

oon 

MOl 

on 

.577 

1  J 

—     6079 

.2419 

-1.098 

-    .963 

6.2 

-    .6208 

7354 

.165 

414 

i    i 

-     7697 

-    .0152 

-     Ill 

-1.400 

54 

-    .3861 

xin:t 

198 

-Ml 

I  r, 

1     X 

-    .8120 
-    .8495 

-    .2561 

-    .137 
-     259 

-    .990 
-    .806 

66 
5.8 

-   .14.M 
+   .0989 

8338 

.8301 

m 

205 

on 

-     nm 

•-'  o 

-    .9180 

-   .5202 

-    .423 

-    .649 

FIG.  32. 


Flo.  33. 


510 


PERIODIC    ORBITS. 


in  dimensions  and  periods;  and  for  (7  =  3.3284  there  is  a  very  close  approach 
to  a  periodic  orbit,  though  one  was  not  actually  found.  The  computations 
indicate  that  the  two  series  of  periodic  orbits  unite  and  disappear  for  some 
value  of  C  slightly  smaller  than  3.3284.  But  there  is  an  orbit  so  nearly 
periodic  for  (7  =  3.3284  that  it  is  included  as  being  very  nearly  that  double 


C  =  3.3284,  Approximate  Double  Periodic  Orbit,  Period  =2X6.  14  =  12.28  (Fig.  34,  page  511). 

1 

X 

y 

x' 

.'/' 

t 

X 

y 

x' 

.'/' 

0 

0 

.3500 

-  .567 

.000 

2.2 

-1.3027 

-  .3950 

-  .497 

.  434 

.05 

-  .0284 

.3499 

-  .571 

-  .003 

2.4 

-1.3953 

-  .2911 

-  .422 

.(MM) 

.10 

-  .0572 

.3496 

-  .583 

-  .006 

2.6 

-  .4691 

-  .1504 

-  .310 

.731 

.18 

-  .0868 

.3492 

-  .603 

-  .010 

2.8 

-  .5178 

+  .0051 

-  .173 

.821 

.20 

-  .1176 

.3486 

-  .632 

-  .016 

3.0 

-  .5375 

.1748 

-  .023 

.  S(>9 

.25 

-  .1501 

.3476 

-  .670 

-  .024 

3.2 

-  .5270 

.3498 

+  .128 

.876 

.30 

-  .1848 

.3461 

-  .717 

-  .036 

3.4 

-  .4869 

.5226 

.270 

.846 

.4 

-  .2624 

.3405 

-  .840 

-  .081 

3.6 

-  .4201 

.6862 

.395 

.785 

.5 

-  .3542 

.3281 

-1.000 

-  .180 

3.8 

-  .3307 

.8350 

.  495 

.701 

.6 

-  .4628 

.3009 

-1.168 

-  .390 

4.0 

-  .2239 

.9655 

.569 

.603 

.7 

-  .5842 

.2444 

-1.226 

-  .768 

4.2 

-1.1050 

.075!) 

.615 

.500 

.8 

-  .6972 

.1460 

-  .979 

-  1  .  174 

4.4 

-  .9795 

.  165S 

.631 

.400 

.9 

-  .7742 

.0206 

-  .568 

-1.273 

4.6 

-  .9517 

.2364 

.638 

.308 

1.0 

-  .8166 

-  .1004 

-  .315 

-1.126 

4.8 

-  .7254 

.2900 

.623 

.  229 

1.1 

-  .8428 

-  .2032 

-  .232 

-  .931 

5.0 

-  .6033 

.3292 

.597 

.166 

1.2 

-  .8658 

-  .2871 

-  .237 

-  .752 

5.2 

-  .4869 

.3575 

.567 

.120 

1.3 

-  .8915 

-  .3542 

-  .280 

-  .595 

5.4 

-  .3763 

.3782 

.  540 

.089 

1.4 

-  .9223 

-  .4066 

-  .335 

-  .455 

5.6 

-  .2705 

.3939 

.520 

.070 

1.5 

-  .9586 

-  .4455 

-  .390 

-  .326 

5.8 

-  .1675 

.4008 

.511 

.060 

1.6 

-1.0000 

-  .4720 

-  .439 

-  .204 

6.0 

-  .0651 

1.4182 

.516 

.055 

1.8 

-1.0954 

-  .4895 

-  .507 

+  .026 

6.2 

+  .0397 

1.4288 

.536 

.051 

2.0 

-1.1996 

-  .4627 

-  .527 

.240 

periodic  orbit  at  which  the  two  single  periodic  orbits  unite  and  disappear. 
It  is  believed  that  in  all  cases  the  computations  covered  so  wide  a  range 
of  initial  conditions  that  no  periodic  orbits  of  the  type  in  question  escaped 
detection. 

The  results  shown  in  the  preceding  tables  (omitting  intermediate 
steps)  were  obtained  by  the  computations,  the  origin  of  coordinates  being  at 
the  center  of  gravity  of  the  finite  bodies. 

246.  Closed  Orbits  of  Ejection  for  Large  Values  of  n- — It  was  shown 
in  Chapter  XV  that  for  small  values  of  n  there  exist  closed  orbits  of  ejection 
from  1  —  ju  for  projections  both  toward  and  from  1  — /z  and  that  their  periods 
reduce  to  2jw  (j=l,  2,  .  .  .  )  for  /x  =  0.  It  was  also  shown  in  §234  that 
these  orbits  can  be  continued,  in  the  analytic  sense,  to  any  value  of  n  unless 
two  of  them  disappear  by  becoming  identical  and  vanishing.  In  order  to 
confirm  this  conclusion  and  to  get  an  idea  of  the  form  of  these  orbits  for 
large  values  of  /*,  63  orbits  of  ejection  were  computed.  It  was  also  desired  to 
discover  orbits  which  are  orbits  of  ejection  from  one  finite  mass  and  of 
collision  with  the  other. 

The  computations  were  all  started  by  means  of  the  series  (36)  of  §228. 
After  the  values  of  x,  y,  x',  and  y'  had  been  determined  for  a  few  small 
values  of  t,  the  computations  were  continued  by  the  ordinary  processes. 


M  N  I  IIK>I> 


I'1.|;|"UH 


.-,11 


In  all  cases  the  infinitesimal   body  \v;is  ejected  from  the  finite  lunly    1  — 
in  the  positive  or  negative  r-ilirection. 


M-i2,  r-  2.242.  doMd  Orbit  of  Ejection  H 

' 

i 

'i 

f 

1 

* 

t 

«" 

0 

AIMHI 

0 

+  oo 

0 

1    1 

M01 

-     ' 

- 

-      - 

lo 

-      - 

-     0266 

-     • 

1241 

.' 

-     .Ml 

•  •'. 

U 

—     1592 

-      O.MJ 

-     549 

1   x 

.--.I 

• 

-   .759 

-   .703 

20 

-     0950 

-    .(Ml 

1    .112 

-     644 

J  u 

1221 

1940 

-   .979 

-    .560 

2~, 

_     (] 

-    .ll.Vi 

!    HM 

- 

-    .0929 

-     5884 

—     163 

-   .367 

-|-     .Oli'.x 

—      l.'i3O 

-.7 

- 

J   1 

—     34O1 

6388 

- 

-     ' 

-    .1934 

•.'HI 

- 

_      ,. 

6386 

- 

+    .139 

III 

HIM 

- 

901 

- 

-      H 

KOO 

- 

:. 

1964 

- 

- 

3  0 

-       ' 

noo 

- 

'. 

2767 

—    .4070 

77.. 

-      MJ 

-     4203 

-   .2931 

-     220 

OOK 

8406 

—     4884 

070 

-     799 

34 

—    ivuvi 

-    0654 

- 

BOO 

I 

1112 

-    '.M.7I 

!CM 

-      7M 

S284 

—     7910 

- 

172 

1 

I'-"  > 

-      • 

IJI 

—     7"»7 

-     9559 

-     4804 

-     4MS 

821 

1   II 

-    : 

-     799 

1  u 

-2  .0209 

-    .1171 

-     162 

ooo 

1    J 

-    .8878 

oil 

- 

42 

-20201 

+     1938 

+     170 

ooo 

„-}$,  C-  2.840,  dosed  Orbit  of  Ejection  (Fig.  36). 

i 

X 

V 

z' 

y' 

t 

i 

V 

•* 

i 

0 

-    .5000 

0 

+  oo 

0 

3  0 

-     9546 

-     019 

-      171 

U) 

-      -Mi:. 

1   .Ml 

-    .408 

3  2 

0000 

-     9930 

-     045 

-     210 

II 

-    .17«0 

-    !0485 

1  214 

-    .506 

3   1 

3710 

-     0390 

-    .086 

-    .250 

20 

-    .121.'. 

-    .0758 

-     580 

3  r, 

-     0930 

-      Ml 

-     290 

.25 

-    .0744 

-     1002 

074 

-     636 

3121 

-     1546 

-     220 

-     325 

.;n 

-    .0335 

-    .1391 

TOO 

-    .677 

4  0 

-     2222 

-     312 

- 

.35 
.40 

+   .0029 
0800 

-    .1737 
-    .2092 

ooo 

-     7i  rj 
-     71.-, 

1   J 
44 

1861 
0007 

-     292K 

8021 

-      ll'i 

- 

- 
- 

OMO 

-    .2806 

.546 

—    .706 

46 

-   .0284 

-     4251 

-     666 

-     280 

1  I  .VI 

-    .34«J2 

IM, 

—     663 

I   x 

-     1717 

-     4761 

- 

-     211 

.7 

1918 

-     4126 

-     604 

.-,  II 

—   .3877 

-     5087 

- 

-      100 

2823 

-    .4698 

-    .540 

5.2 

-    5232 

-     5165 

-     969 

+     033 

9 

1870 

-    .5208 

821 

-    .481 

54 

-     7234 

-     4938 

-1 

ins 

0 
2 
4 
6 
B 

2  II 

J    1 
26 

J'.Mi| 

8403 
8929 

-    .5662 
-    .6424 
-    .7028 
-    .7509 
-    .7894 
-   .8208 
-   .8474 
-    .8716 
-    .8959 

174 
102 

on 

022 

006 

ooo 

(KK) 
-    .001 

-    .427 
-     338 
-   .269 
-    .214 
-      173 
-     143 
-    .125 
-      1  !'.» 
-    .126 

56 
6.0 
6  4 

7  u 
7  2 

-   .9319 
-     1408 
-     3417 
-     5249 

-     • 
-      • 
-     SX34 
-     9164 
-     S976 

-     4359 
-     3395 
-     2033 

- 
-      M7J 
-     :>764 
-      3130 

-1 
-    .968 

-   .857 

- 
-     .Mil 

- 
- 
+ 

780 

HO 
188 

."7 

-.•' 

2.8 

3923 

-    .9228 

-   .005 

-    .11.-, 

FIG.  35. 


512 


PERIODIC   ORBITS. 


Any  orbit  which  intersects  the  x-axis  perpendicularly  is  symmetrical 
with  respect  to  the  .r-axis.  Hence,  it  follows  that  if  one  of  these  orbits  of 
ejection  intersects  the  .r-axis  perpendicularly,  then  it  is  a  closed  orbit  of  ejec- 
tion of  the  type  treated  in  Chapter  XV. 

Computations  were  first  made  for  /*  =  f  to  discover  orbits  of  the  typo 
characterized  by  j=l  with  ejection  toward  fj,  and  shown  in  Fig.  15.  It  was 
proved  in  Chapter  XV  that  such  an  orbit  exists  for  small  values  of  n  and  that 
its  period  is  approximately  2ir.  Such  an  orbit  was  found  for  ju  =  |,  but  its 
period  was  about  8.  Another  orbit,  also  of  a  similar  type,  was  discovered 
whose  period  was  about  14.  One  of  these  orbits  is  undoubtedly  the  limit, 
for  decreasing  values  of  C,  of  the  oscillating  satellite  about  the  collinear 
equilibrium  point,  as  Burrau's  calculations  have  indicated.  The  value  of 
C  corresponding  to  the  equilibrium  point  for  M  =  |  is  about  3.46,  and  the 
values  of  C  for  these  orbits  are  2.24  and  2.84.  The  question  arises  regarding 
the  origin  of  the  other  orbit  of  this  type.  It  is  probably  the  limiting  form 
of  a  periodic  orbit  about  1—  n  consisting  of  a  double  loop  and  having  a 
double  point  on  the  x-axis.  Such  orbits  were  treated  by  Poincare  in  Les 
Methodes  Nouvelles  de  la  Mecanique  Celeste,  Chapter  XXXI.  The  ana- 
lytic continuation  of  the  former  beyond  the  ejectional  form  for  decreasing 
values  of  C  is  also  a  periodic  orbit  with  two  loops.  For  greater  or  smaller 
values  of  C  the  latter  will  have  also  the  character  of  an  oscillating  satellite, 
but  it  can  not  reduce  to  the  equilibrium  point  because  there  is  only  one  orbit 
of  this  type.  The  results  for  ^  —  ^  are  given  in  the  tables  of  page  511. 

For  M  =  |  similar  results  were  found.  Since  orbits  of  this  type  have 
not  been  computed  heretofore,  the  results  for  the  four  orbits  will  be  given 
for  enough  values  of  t  to  exhibit  their  properties. 

The  corresponding  results  for  /x  =  y>  with  ejections  from  1— ju  =  |,  are 
given  in  the  following  tables : 


M=4/5,  C=  2.696,  Closed  Orbit  of  Ejection  (Fig.  37,  page  513) 

I 

X 

V 

x' 

V 

1 

X 

y 

x' 

V1 

0 

-  .8000 

0 

+  co 

0 

2.0 

.5307 

-1.0537 

-  .130 

-  .605 

.10 

-  .6021 

-  .0198 

i.252 

-  .324 

2.2 

.4863 

-1.1762 

-  .314 

-  .616 

.15 

-  .5447 

-  .0383 

1.062 

-  .413 

2.4 

.4051 

-1.2978 

-  .497 

-  ..•)<« 

.20 

-  .4947 

-  .0609 

.945 

-  .489 

2.6 

.2892 

-1.4112 

-  .672 

-  .:,:!.-, 

.25 

-  .4495 

-  .0870 

.870 

-  .554 

2.8 

.1384 

-1.5094 

-  .834 

-  .440 

.30 

-  .4073 

-  .1162 

.822 

-  .610 

3.0 

-  .0429 

-1.5851 

-  .975 

-  .311 

.35 

-  .3669 

-  .1478 

.796 

-  .657 

3.2 

-  .2498 

-1.6316 

-1.090 

-  .150 

.40 

-  .3273 

-  .1816 

.788 

-  .694 

3.4 

-  .4766 

-1.6430 

-1.172 

+  .040 

.5 

-  .2477 

-  .2535 

.811 

-  .735 

3.6 

-  .7158 

-1.6142 

-1.214 

.250 

.6 

-  .1642 

-  .3270 

.861 

-  .725 

3.8 

-  .9591 

-1.5419 

-1.212 

.474 

.7 

-  .0758 

-  .3969 

.904 

-  .668 

4.0 

-1.1972 

-1.4244 

-1.162 

.702 

.8 

+  .0154 

-  .4596 

.912 

-  .585 

4.2 

-1.4205 

-1.2617 

-1.063 

.924 

.9 

.1052 

-  .5140 

.878 

-  .506 

4.4 

-1.6189 

-1.0561 

-  .915 

.128 

1.0 

.1899 

-  .5615 

.810 

-  .449 

4.6 

-1.7829 

-  .8125 

-  .720 

.303 

1.2 

.3343 

-  .6464 

.628 

-  .415 

4.8 

-1.9041 

-  .5384 

-  .487 

.433 

1.4 

.4404 

-  .7321 

.434 

-  .450 

5.0 

-1.9758 

-  .2429 

-  .226 

.513 

1.6 

.5081 

-  .8282 

.244 

-  .511 

5.2 

-1.9936 

+  .0627 

+  .049 

.533 

1.8 

.5382 

-  .9361 

.056 

-  .567 

VI  1IK-1-    (>K     I'KUIODK      MKIIII-. 


513 


,•-4/5,  r-2965.  Clonl  <M.ii  ,rf  Kjrrtioo 

t 

r 

V 

*' 

s 

1 

z 

1 

•• 

»' 

1) 

-      .Mill 

II 

+  00 

0 

-     '.HI., 

- 

- 

10 

0009 

0193 

1     ll.s 

7  jr.1           TII-I 

-     li-i 

II- 

l.'l 

• 

-    .u:pi 

- 

11 

-     'Jit 

-     444 

n 

-      .Mli: 

0.-.77 

- 

- 

- 

1713 

-     .(Nil, 

746 

- 

-       . 

-     II, 

- 

I.Crfi       -      1(17'.' 

- 

- 

'   .' 

-       |. 

-     i:«n 

Mi 

- 

0 

3561 

an 

-   .609 

in 

-       ! 

-   .807 

J 

1041 

- 

-     334 

..-. 

- 

-    -JJ-J 

MI 

-   .688 

i 

MM 

Ml 

_     sit. 

-     2*5 

i 

M4S 

-    I 

-    833 

', 

-  .ion 

-     8047 

-  .8W 

- 

.7 

-    .177:: 

- 

.,'.; 

-     599 

1 

-     2917 

-     • 

-     96K 

- 

1 

-     III.M 

-      I10-.I 

Tin 

-    .538 

-    .4807 

M7I 

-1  in1' 

+     Ki.' 

0 

—    nj-i-i 

-    »l-»" 

m 

-   .480 

-    6977 

- 

-1  n|7 

it 

+    .017:. 

-    ..VIJ7 

.776 

—    .378 

:,  I 

-     9077 

-    ;,i  i- 

-1     IMS 

IJI 

•2 

IW1 

-      .V.-J7 

-    .332 

5.8 

-    m:. 

-      41.1 

-1   (IK. 

M 

1 

::.;.-,.,             VKC 

-     Us 

-      1 

- 

M 

'i 

I.M.; 

tat 

-     I.'", 

60 

mi 

-     1075 

- 

•n 

1 

:,i:,l            n.vji 

n« 

-     Ill 

'.  J 

-      WW» 

-     9057 

- 

UK! 

J  n 

lil'.Ki                 lis|| 

-       |s| 

64 

-    .774X 

-    .6762 

-    :.n 

MM 

f-746      -    .7-2M 

- 

6.6 

-     - 

-    .4249 

- 

2«H 

J    1 

7133 

-      7771 

148 

- 

-     8103 

-     1M9 

- 

ma 

-      - 

- 

7  ii 

-   .9139 

+     1102 

4-  .OW 

( '(imputation.-  were  also  made  in  which  the  ejection  wii*  in  the  n<>KHtivc 
direction  from  1  —  p.  One  periodic  orbit  of  the  type  characterized  by  ./—I, 
Fiji.  15,  was  discovered,  and  its  coordinates  are  pven  in  the  following  table: 


„_!.,,  C-  1.8224,  Clowcl  Orbit  of  Kjrrti.m  (Fig  39). 

< 

X 

1 

«• 

r- 

1 

i 

9 

z' 

< 

0 

-       Muni 

0 

—  00 

0 

.3 

-1  0232 

4027 

902 

.10 

-      7 

0873 

-1.736 

IIS 

I 

-    .9251 

.5296 

050 

\     1  '>- 

.  K. 

-    .KMC, 

.0580 

-1.457 

.-.77 

5 

-     .Klls 

.8337 

208 

1    IX,  J 

-       '.U  '.K! 

"s  17 

-1.264 

J 

-   .6843 

7344 

Ml 

-    .97*4 

1218 

-1.111 

7-1 

.7 

-     5439 

Mfl 

Bl 

M 

-     0306 

-    .979 

-7H 

- 

-    .3919 

«ttl 

•n 

.'•K 

-    .0765 

-      .Mil 

.947 

2.0 

-    .0595 

711 

'4 

-     lir.s 

-     7.M 

Ol'i 

••  'I 

+   .2993 

2  03X7 

-    .1816 

-    .547 

189 

24 

MM 

J  0011 

-1- 

- 

'j. 

-     2265 

1827 

991 

26 

1  0331 

1    ss7'l 

77- 

- 

7 

-   .ivji 

-   !l64 

Ml 

J  - 

1    .!717 

1.7009 

—i  in 

-     2593 

:«2 

:i  it 

1  6777 

1     UM 

:{93 

-1   436 

9 

-     2478 

907 

3  2 

1.9271 

1   1303 

-1  7(M 

10 

1    1 
1  i' 

-    .2180 

-      17(Ct 

1   OOWi 

1  1 

1  2780 

7:!7 

Ml 
MB 

295 

:»  4 
36 
3.8 

2  217tt 
2  2445 

7..S-, 
3755 
-     0315 

:Ws 
—    .070 

-1  901 

-2  01.-, 
-2  039 

9  Am 


T-A.i 


I 


• 


514 


PERIODIC    ORBITS. 


If  n  were  zero  and  the  infinitesimal  body  were  ejected  from  1  — M  either 
toward  or  from  ^  in  such  a  way  that  its  period  would  be  TT,  the  orbits  described 
in  rotating  axes  would  consist  of  two  parts  symmetrical  with  respect  to  the 
x-axis,  as  shown  in  Figs.  40,  a,  and  41,  a.  These  curves  are  the  limits  sepa- 
rating two  types  of  periodic  orbits  (for  M  =  0)  in  the  rotating  plane,  as  is 
shown  in  Figs.  40,  b,  and  41,  c.  As  n  increases  a  dissymmetry  develops 
with  respect  to  the  line  through  1  —  M  perpendicular  to  the  x-axis.  Suppose 
the  orbits  are  followed  as  yu  increases  in  such  a  way  that  they  shall  remain 
orbits  of  ejection  in  one  way  or  the  other.  Then  orbits  of  the  type  Fig. 
40,  a,  will  go  into  types  having  some  of  the  characteristics  of  both  types  a 
and  c  of  Fig.  41.  That  they  partake  of  the  characteristics  of  type  c  instead 
of  those  of  type  b  was  proved  by  computations  for  both  n  =  %  and  n=  i. 

The  two  following  tables  give  the  results  for  ejection  from  1—n  toward 
fj.,  with  n  having  the  values  \  and  -|-  The  first  orbit  lacks  a  little  of  being 
closed,  but  an  exactly  closed  orbit  exists  between  this  one  and  the  one  which 
was  computed  for  (7  =  3.478. 


M  =  Ji  0  =  3.489,  Closed  Orbit  of  Ejection  (Fig.  42). 

t 

X 

V 

x' 

y' 

t 

X 

y 

x' 

.'/' 

0 

-.5000 

0 

+   CO 

0 

.7 

-.0225 

-.2933 

-    .038 

-.229 

.1 

-.2559 

-  .0244 

1.353 

-.380 

.8 

-.0290 

-.3090 

-    .088 

-.089 

.15 

-.1972 

-  .0455 

1.021 

-.456 

.9 

-.0393 

-.31  OS 

-    .114 

.054 

.20 

-.1521 

-.0697 

.793 

-.506 

.0 

-.0511 

-.2980 

-    .  11<> 

.203 

.25 

-.1169 

-.0958 

.621 

-.533 

.1 

-.0625 

-.2703 

-    .107 

.353 

.30 

-.0894 

-  .  1227 

.486 

-.543 

.2 

-.0721 

-.2276 

-    .083 

.499 

.35 

-.0680 

-  .  1498 

.375 

-.537 

.3 

-.0790 

-  .  1708 

-    .055 

.634 

.40 

-  .0516 

-  .  1762 

.283 

-.518 

.4 

-  .0833 

-  .  1018 

-    .033 

.741 

.5 

-  .0308 

-.2248 

.141 

-.448 

.5 

-.0860 

-  .0244 

-    .022 

.796 

.6 

-  .0222 

-.2647 

.037 

-.346 

1.55 

-  .0871 

+  .0154 

-    .022 

.794 

Six  orbits  of  a  similar  type  were  computed  for  M  =  |-     The  following 
approximately  closed  orbit  was  obtained: 


-YMIIl  -I- 


I'KIM  .......  .1:1111-. 


515 


M-4/5,  ('  -3.5927,  OOM-.I  ori.it  ,.:      .                       «). 

( 

z 

V 

-r- 

>/• 

« 

i 

1 

/• 

•' 

ii 

-.SHOO 

0 

+  00 

ii 

7 

-  •• 

-   i 

-  036 

-  097 

1 

-   0177 

'..:,7 

s 

-   477'. 

-    1"'.7 

-  056 

+  fllo 

u 

-     • 

_   n 

7(0 

-   1 

.t 

-   4X34 

- 

-  000 

ISO 

•Jii 

5611 

0497 

_    . 

1   II 

-  4892 

-    17H. 

BO 

-.0077 

in 

1 

-  4937 

-   1433 

-   03X 

:«i 

-  .MIMI 

-.368 

.' 

-.4977 

-.1051 

-  094 

-    i 

-  nut 

- 

-  4987 

-  0.188 

-  010 

i 

-    ; 

-  r.Ms 

1..7 

-  :u«.i 

1 

-..VXK 

-  m:t 

-   191 

.."> 

1711 

-  i  .-,•.-. 

069 

m 

-   ! 

-1-  0 

-    •-'!> 

-  »rr. 

-    I7i,7 

mil 

-.195 

Computation.-  wen-  made  for /*=  1  in  which  the  ejections  were  from  1—  n 
in  tin-  direction  o|)|>osite  to  p.  One  dosed  orbit  corresponding  to  j  =  |  was 
t'ound,  the  results  for  which  are  contained  in  the  following  table.  Corre- 
sponding computations  were  not  made 


,1-1.  .1       .'7;.  n.«iHM>itof  Kjivii,,n  iKin    II 

t 

X 

V 

z' 

»' 

t 

z 

* 

z- 

v' 

0 

-    .SOOO 

0 

—  oo 

0 

4 

-    .7711 

Ml 

.566 

.162 

-1 

-     7 

-1.509 

.415 

5 

-    .7IM 

u 

-     - 

0490 

-1   •-!!.-. 

:>i:< 

1 

-    .6644 

.8156 

:,is 

-    .067 

.11 

-    .8800 

.0766 

-    .996 

592 

7 

-     6151 

HMfl 

465 

-    .163 

M 

-    .9254 

1117s 

-    .823 

6M 

1 

-     5719 

M 

-    .383 

-    .9628 

Ills 

-     1,77 

71  W 

'.1 

-     5367 

307 

-    .367 

-    .9934 

1799 

-    .549 

7111 

20 

-     .Mil 

71J1 

- 

III 

-       Ills!) 

21M 

-    .434 

767 

•.'  1 

-    .4987 

am 

m 

-   .633 

.5 

-    0.-.K) 

J'UH 

- 

.795 

22 

-    .4951 

MM 

-     052 

-   .615 

.6 

-     .tNVtf 

_          II.-.N 

.793 

•2  :t 

-    .5076 

-    an 

- 

.7 

-    .(Hi3Ti 

1517 

+    out 

766 

•J  4 

-    .5356 

MM 

-    :«ii 

-      SM 

8 

-      (H7i. 

233 

717 

- 

- 

- 

.9 

-      (ll'.'v 

:tti 

651 

26 

-     6374 

MM 

-     619 

-      JII 

1  (i 

-    .9821 

.419 

.-,7(1 

27 

-     6944 

1214 

-      u:. 

-      • 

1    1 

-    .9366 

7081 

.487 

.478 

J  72.'. 

-     7<MO 

ii7-7 

-  ,3i« 

- 

1   •_' 

-     8854 

7:.  Hi 

BM 

J  7.^1 

-     7100 

-       IV, 

-    .808 

-     6 

B01 

•J71 

.'  77:. 

-     71  Hi 

-   OHJ 

+   .026 

-     800 

IT" 


\ 


I...    II 


247.  Orbits  of  Ejection  from  1-M  and  Collision  with  M-—  The  motion 
of  the  infinitesimal  body  from  ejection  may  be  followed  by  means  of  the 
series  (36)  of  §228.  The  motion  of  the  infinitesimal  body  into  a  collision 
with  the  same  or  the  other  finite  mass  can  not  be  followed  by  numerical 
processes.  But  a  collision  with  the  second  finite  mass  can  be  established  in 


516 


PERIODIC    ORBITS. 


certain  cases  by  making  use  of  properties  of  symmetry.  Any  orbit  which, 
for  M  =  5>  intersects  the  y-axis  perpendicularly  is  symmetrical  with  respect 
to  the  t/-axis.  Hence  it  follows  that  if,  for  n  =  %,  an  orbit  of  ejection  from 
1— M  intersects  the  7/-axis  perpendicularly,  then  it  has  a  symmetrical  col- 
lision with  the  second  finite  mass. 

After  14  computations  had  been  made,  an  orbit  of  ejection  from  1— /x 
and  collision  with  /z  (ju  =  |)  was  discovered  in  which  the  ejection  was  toward 
M  and  in  which  the  collision  took  place  without  the  infinitesimal  having 
encircled,  in  the  rotating  plane,  either  1  —  n  or  n.  The  following  table  gives 
the  results  at  a  considerable  number  of  intervals  from  the  time  of  ejection  of 
the  infinitesimal  body  from  1  —  n  until  it  crossed  the  line  x  =  0 : 


it=lA,  0  =  3.4174,  Orbit  of  Ejection  and  Collision  (Fig.  45,  page  517). 

t 

X 

y 

x' 

y' 

t 

x 

y 

x' 

y' 

0 

-.5000 

0 

+   00 

0 

.  55 

-.0082 

-.2549 

.138 

-.440 

.10 

-.2540 

-.0240 

1.371 

-.383 

.(JO 

-.000.-) 

-.2751 

.094 

-.391 

.15 

-  .  1949 

-.0459 

1.042 

-.402 

.65 

+  .0012 

-.2933 

.058 

-.338 

.20 

-.1487 

-.0704 

.818 

-.514 

.70 

.0033 

-.3088 

.028 

-.281 

.25 

-.1122 

-.0970 

.650 

-.545 

.75 

.0040 

-.3214 

.004 

-.222 

.30 

-.0831 

-  .  1247 

.518 

-.559 

.80 

.0038 

-.3310 

-    .014 

-.161 

.35 

-.0600 

-  .  1527 

.412 

-.557 

.85 

.0028 

-.3375 

-    .026 

-.099 

.40 

-.0417 

-  .  1803 

.324 

-.542 

.90 

.0013 

-.3409 

-    .032 

-.037 

.45 

-.0274 

-.2068 

.251 

-.516 

.95 

-.0003 

-.3411 

-    .032 

+  .027 

.50 

-.0104 

-.2318 

.190 

-.482 

Another  somewhat  similar,  but  larger,  orbit  of  ejection  from  1— /x  and 
collision  with  /x  was  found,  after  a  number  of  computations,  in  which  the 
ejection  was  toward  n.  Part  of  the  results  of  the  computation  are  given  in 
the  following  table : 


M  =  K,  C  =  2.739,  Orbit  of  Ejection  and  Collision. 

I 

x 

y 

x' 

y' 

t 

x 

y 

x' 

y' 

0 

-.5000 

0 

+  °° 

0 

1.4 

.3941 

-  .7683 

.032 

-.379 

.10 

-.2428 

-  .0258 

1.537 

-.412 

1.5 

.3947 

-  .8051 

-  .018 

-.357 

.15 

-  .  1739 

-  .0490 

1.242 

-.514 

1.6 

.3906 

-  .8398 

-  .063 

-.330 

.20 

-.1109 

-  .0707 

1.048 

-.592 

1.7 

.3822 

-  .8723 

-  .105 

-.315 

.25 

-.0682 

-  .1079 

.910 

-.652 

1.8 

.3698 

-  .9028 

-  .142 

-.295 

.30 

-  .0254 

-  .1417 

.808 

-.096 

1.9 

.3539 

-  .9313 

-  .176 

-.274 

.35 

+  .0131 

-  .1773 

.733 

-.720 

2.0 

.3348 

-  .9577 

-  .206 

-.253 

.40 

.0483 

-  .2140 

.077 

-.742 

2.1 

.3129 

-  .9819 

-  .233 

-.232 

.45 

.0810 

-  .2512 

.836 

-.745 

2.2 

.2884 

-1.0041 

-  .257 

-.211 

.50 

.1119 

-  .2883 

.000 

-.737 

2.3 

.2616 

-1.0241 

-  .278 

-.189 

.68 

.1412 

-  .3248 

.570 

-.720 

2.4 

.2328 

-1.0419 

-  .296 

-.107 

.6 

.  1690 

-  .3602 

.542 

-.697 

2.5 

.2024 

-1.0575 

-  .312 

-.145 

.7 

.2202 

-  .4273 

.482 

-.042 

2.6 

.1705 

-1.0709 

-  .326 

-.122 

.8 

.2651 

-  .4880 

.410 

-.580 

2.7 

.1373 

-1.0820 

-  .337 

-.100 

.9 

.3033 

-  .5440 

.347 

-.530 

2.8 

.1031 

-1.0908 

-  .346 

-.077 

1.0 

.3345 

-  .5960 

.277 

-.493 

2.9 

.0682 

-1.0973 

-  .353 

-  .053 

1.1 

.3588 

-  .0435 

.210 

-.458 

3.0 

.0326 

-1.1014 

-  .357 

-.030 

1.2 

.3706 

-  .6878 

.146 

-.428 

3.1 

-.0032 

-1.1032 

-  .360 

-.006 

1.3 

.3882 

-  .7293 

.087 

-.402 

M  \  I  IU.-I- 


'l   Kl  ......      iillHIIS. 


517 


This  orbit  has  a  luu|.  about  the  equilateral  triangular  point        It  follou.- 
that  then-  an-  twu  families  of  |>eriodic  orhit>  <,f  the  type.-  >ho\\n  in  I  IM     |i; 
Ten  urhits  were  computed  in  an  attempt  to  find  one  of  tlu-in.     The  diffi- 
ciillics  of  making  the  calculations  when  the  infinitesimal  body  \V:L-  near  one 
of  the  finite  bodies  were  >o  Kreat  that  wide  depart urcs  from  the  orl.n 
ejection   had   to  he  attempted.      Indications  of  such  jM-riodic   orbits   \ 
obtained,  but  none  \\a>  actually  found. 


Via.  45.  Kin.  46. 

248.  Proof  of  the  Existence  of  an  Infinite  Number  of  Closed  Orbits  of 
Ejection  and  of  Orbits  of  Ejection  and  Collision  when  n  =  \. — It  was  proved 
in  Chapter  XV  that  when  n  is  sufficiently  small  there  are  infinitely  many 
closed  orbits  of  ejection,  and  reasons  were  given  for  believing  that  the-. 
orbits  persist  for  all  values  of  /u-  The  question  of  the  existence  of  orbit >  of 
ejection  and  collision  was  not  considered. 


Fio.  47.  Fio.  48. 

It  will  now  be  shown  that  there  are  infinitely  many  closed  orbits  of  ejec- 
tion, and  of  ejection  and  collision,  for  M  =  i-  The  differential  equations  of 
motion  in  fixed  rectangular  axes  with  the  origin  at  the  center  of  gravity  <  »f 
the  system  are 

rVl),       ,,s 


dP  r,'  r,«  dP 

where,  if  the  finite  bodies  are  on  the  r-axis  at 


in*.     (69) 


518  PERIODIC    ORBITS. 

Now  let  x  =  r  cos  6,  and  y  —  r  sin  6,  after  which  equations  (68)  become 

tfr    /de\~   ari    11    ifi    11 

jJI-Mjrl  —  3|-lH  —  ilr  —  2   ~i  --  i     COS  (0—  Oi 

d£-       \d<y      2|_ri8     fj»J       *[ria     'V3J 

df  d8\     !/l      IN     . 

d«(^rKn'~tf)r8m('~0' 
where 


i-r  cos(0-0-      (71) 
Suppose  it  is  known  that,  at  t  =  T, 

rS2,     §>().  (72) 


always  positive,  it  follows  from  the  first  of  (70)  that 

dV     -  (r+il 
&•>     (r-i)*' 

The  integral  of  this  inequality  gives 


dr 
Suppose  K>0  and  that  ^>0  at  1  =  T.     Then  F(r)  will  always  exceed  A'  in 

value  and  r  will  become  infinite  with  t. 

Computations  were  made  in  which  the  infinitesimal  body  was  ejected 
from  1  —  M  both  toward  and  from  /j.  with  initial  conditions  corresponding  to 

dr 
K>0,  and  they  were  both  followed  until  r>2  with  ^  positive.     Hence  in 

both  cases  the  infinitesimal  body  would  recede  to  infinity.  Moreover,  it 
follows  from  the  second  equation  of  (70)  that,  when  referred  to  rotating 
axes,  the  infinitesimal  body  revolves  infinitely  many  times  about  the  finite 
bodies,  and  its  distance  from  the  origin  continually  increases. 

Now  consider,  for  example,  a  closed  orbit  of  ejection  from  1—  M,  for 
M  =  5,  in  which  the  infinitesimal  body  makes  at  least  one  circuit  about  the 
finite  body  ju.  Its  orbit  therefore  crosses  the  z-axis  perpendicularly  exactly 
once,  but  it  does  not  cross  the  ?/-axis  perpendicularly.  Moreover,  it  follows 
from  the  symmetry  of  the  orbit  with  respect  to  the  x-axis,  that  if  it  crosses 
the  i/-axis  in  the  first  half  of  the  orbit  at  an  angle  T/2  +  a,  then  in  the  second 
half  it  crosses  the  i/-axis  at  the  angle  ir/2  —  a.  One  or  the  other  of  these  angles 
is  less  than  ir/2.  Suppose  it  is  the  latter.  Now  suppose  the  initial  conditions 
of  ejection  are  changed  so  as  to  increase  K.  The  orbit  will  cease  to  be  a 
closed  orbit  of  ejection  and  will  tend  toward  one  which  winds  out  to  infinity 
with  continually  increasing  r.  The  angles  at  which  the  orbit  crosses  the 
axes  are  continuous  functions  of  the  parameter  defining  the  initial  conditions, 
as  K  for  example.  Hence  the  intersection  with  the  ^/-axis  which  was  at  the 


M  viiiK-i-    i.i     i-i  iMdim     ORB1  .")!'.» 

angle  T  2  —  o  and  less  than  TT  2  for  tin-  c|o>ed  orbit  of  ejection  will  \H-  exactly 
perpendicular  fur  a  certain  value  of  A'.  Tin-  <>rl>it  \\ill  be  therefore  an  orbit 
of  ejection  from  1  —  ^  and  collision  with  /j. 

Now  consider  an  orbit  of  ejection  from  1  -M  and  collision  with  p.  cross- 
ing the  .r-axis  at  least  once.  From  t  he  s\  ninietn of  these  orbits  with  reaped 
to  the  //-axis  it  follows  that  they  cm—  the  X-tOU  an  even  number  of  time* 
and  that  if  such  an  i  rbit  crosses  the  .r-a\i-  once  at  an  angle  T  '2+J,  win 

positive  quantity,  then  it  also  crosses  it  at  an  angle  of  T  2  —  ^.  If  the 
initial  conditions  are  so  changed  as  to  increase  the  constant  K,  the  orbit 
ceases  to  be  an  orbit  of  ejection  and  collision,  the  angle  corresjxmding  to 
TT  '2  —  ,i  eventually  becomes  greater  than  r  2,  and  therefore.  Miice  it  i-  ;i 
continuous  function  of  A',  there  is  at  lea>t  one  value  of  A'  for  which  it  i- 
exactly  TT  I*.  Such  an  orbit  is  a  closed  orbit  of  ejection. 

It  follows  from  this  discussion  that  if,  for  any  value  of  A",  there  is  a  closed 
orbit  of  ejection,  then  for  some  larger  value  of  A",  oomspoodmg  to  a  smaller 
value  of  the  Jacohian  constant  (',  there  is  an  orbit  of  ejection  and  collision; 
and  that  if.  for  any  value  of  A",  there  is  an  orbit  of  ejection  and  collision, 
then  for  some  larger  value  of  A.  corresponding  to  some  smaller  value  of  the 
Jacobian  constant  (',  there  is  a  closed  orbit  of  ejection.  Hence,  for  1  — M*  J, 
there  are  infinitely  many  closed  orbits  of  ejection  and  collision.  They  are 
all  distinct  because  they  have  distinct  values  of  A'.  And  since  it  has  i 
shown  that,  when  K>0  for  an  orbit  of  ejection,  r  increases  continuously  to 
infinity,  it  follows  that  the  infinite  sets  of  values  of  A'  corres|xmding  to  these 
classes  of  orbits  are  bounded.  The  orbits  may  be  characterized  by  the  num- 
ber of  times  they  cross  the  //-axis.  For  ejections  in  both  the  positive  and  the 
negative  direction  there  are  closed  orbits  of  ejection  from  each  of  the 
finite  bodies,  and  also  orbits  of  ejection  from  one  and  collision  with  the 
other,  which  eross  the  //-axis  2(2/-f  1)  times,  j  =  Q,  1,  2,  ... 

249.  On  the  Evolution  of  Periodic  Orbits  about  Equilibrium  Points.— 

The  evolution  of  the  jwriodic  orbits  about   the  equilibrium  points  un  and 
(c)  which  are  on  the  .r-axi>  and  not   between  1—  /*  and  n  was  traced,  for 
decreasing  values  of  C.  by  Hurrau's  computations  from  small  ovals  to  the 
ejectional  form.     For  M  =  $  they  are  shown  in    Fig.  'M.  and  for  /*  = 
Fig.  37.     Heyond  these  forms  they  have  loops  about  1  -p. 

The  periodic  orbits  about  the  equilibrium  ]>oint  </i)  between  1-M 
and  M  undergo  corresponding  evolutions.  In  the  case  /i  =  J  as  the  orbit, 
for  decreasing  values  of  T.  becomes  an  orbit  of  ejection  from  one  body 
it  becomes  an  orbit  of  collision  with  the  other.  This  limiting  form  i- 
shown  in  Fig.  !.">.  It  intersects  the  //-axis  si\  times,  twice  ]>er|)cndicularly. 
Heyond  this  form  it  has  loops  about  the  finite  bodies,  and  the  motion  in 
these  loops  is  in  the  retrograde  direction.  With  decreasing  values  of  C 
these  loops  probably  enlarge,  the  loop  about  each  body  eventually  In-com- 
ing an  orbit  of  collision  with  the  other  IMM!V.  In  this  MM  th«-  orbit  of 


520  PERIODIC    ORBITS. 

ejection  and  collision  intersects  the  t/-axis  six  times,  two  of  the  intersec- 
tions being  perpendicular. 

As  C  decreases,  the  ejectional  and  colliskmal  form  passes  into  a  loop 
about  the  second  body,  which  in  turn  expands  and  becomes  an  ejectional 
and  collisional  form  with  respect  to  the  first  body.  In  this  manner  the 
loops  of  the  periodic  orbit,  with  decreasing  values  of  d,  pass  through  ejec- 
tional and  collisional  forms,  first  with  one  finite  body  and  then  with  the  other, 
in  a  never-ending  series,  the  ejectional  and  collisional  forms  being  those  shown 
to  exist,  for  M  =  i>  in  §248,  in  which  the  ejections  from  each  body  are  in  the 
direction  away  from  the  other.  They  are  characterized  by  the  fact  that  they 
cross  the  ?/-axis  2(2/+l)  times,  j  =  Q,  1,  2,  .  .  . 

If  the  finite  masses  are  unequal  the  evolution  of  the  periodic  orbits  is  in 
a  general  way  similar,  except  that  the  ejectional  forms  for  the  two  masses  do 
not  occur  for  the  same  values  of  C. 

Periodic  orbits  about  the  equilateral  triangle  equilibrium  points  have 
been  shown  in  Figs.  30  and  34.  With  decreasing  values  of  C  they  probably 
increase  in  size  and  pass  through  ejectional  forms,  but  ejectional  forms  in 
which  the  direction  of  ejection  is  not  along  the  .r-axis.  Consequently  they 
can  not  be  discovered  by  numerical  processes.  If  this  conjecture  is  cor- 
rect, for  still  smaller  values  of  C  they  possess  loops  about  the  finite  bodies, 
and  for  still  smaller  values  of  C  the  loop  about  each  of  the  finite  bodies 
may  pass  through  an  ejectional  form  with  the  other.  There  is,  however, 
no  evidence  to  guide  conjectures. 

250.  On  the  Evolution  of  Direct  Periodic  Satellite  Orbits. — There  are 
three  direct  periodic  satellite  orbits,  two  of  which  are  complex  for  large 
values  of  the  Jacobian  constant,  but  all  of  which  are  real  for  smaller  values 
of  C.  Suppose  the  orbits  about  the  finite  body  n  are  under  consideration. 
With  decreasing  value  of  C  they  can  pass  through  ejectional  forms.  In  fact, 
Darwin's  computations  showed  that  two  of  them  were  approaching  such 
forms,  one  by  approaching  p.  from  the  positive  direction  and  the  other  by 
approaching  it  from  the  negative  direction. 

The  motion  of  the  infinitesimal  body  when  it  is  near  collision  is  nearly 
the  same  as  it  would  be  if  the  mass  of  the  second  body  were  zero.  Conse- 
quently its  properties  can  be  inferred  from  a  consideration  of  the  motion  in 
the  neighborhood  of  an  ejection  in  the  problem  of  two  bodies  referred  to 
rotating  axes.  It  is  clear  that  if  the  ejection  is  in  the  positive  direction,  the 
curve  near  the  point  of  ejection  lies  on  the  negative  side  of  the  x-axis,  while 
if  the  ejection  is  in  the  negative  direction  the  curve  near  the  point  of  ejection 
lies  on  the  positive  side  of  the  x-axis.  When  the  ejection  is  in  the  positive 
direction,  the  two  families  of  periodic  orbits  which  are  near  the  ejectional 
orbit  both  intersect  the  z-axis  in  the  negative  direction  from  the  point  of 
ejection,  and  the  small  complete  loop  about  the  point  of  ejection  is  then 
described  in  the  retrograde  direction,  while  the  partial  loop  is  described  in  the 


MMIH-.sIS    oK    l-KKIolMC    oldUl-v  .YJ  1 

positive  direction;  while  if  tin-  ejection  is  in  the  negatm-  direction,  the  t\\.. 
families  of  periodic  orltits  which  arc  near  the  ejections!  orbit  both  inleraact 
ilie  .r-axis  in  the  positive  direction  from  the  point  of  ejection,  l.nt  in  this  case 
the  .small  loops  arc  hoth  described  in  the  same  direction-  a-  in  the  other  case. 

The  .r-a\i-  roiisi>t-  of  three  part-.  \  i/.  that  extending  from  -  -c  to  the 
1-osiiion  of  1  —IJ.  that  extending  from  1  -^  to  n,  inul  that  extending  from  n 
to  -foo .  If  a  periodic  oriiit  inter-ect-  the  j--axis  |x-r|M-ndicularly  in  anyone 
of  the-e  three  parts  before  it  goes  through  an  eject  imial  form.  I  hen  it  will 
also  intersect  the  .r-a\is  pcr|>eiidicul;irly  in  the  same  part  after  it  p:isM8 
through  the  ejectional  form.  Moreover,  the  branches  of  a  closed  orbit  of 
ejection  extend  from  the  finite  body  with  which  there  is  collision  in  tin- 
direct  ion  opposite  to  that  in  which  the  neighboring  jx-riodic  orbit  intersects 
the  .c-axis  perpendicularly. 

In  the  case  of  the  direct  periodic  orbit  about  1  —  n  which  enlarge-  in  t  he 
po-itive  direction  and  approaches  1— ^  from  the  negative  direction,  the 
ejection  is  in  the  positive  direction,  the  collision  is  in  the  negative  direction, 
and  the  orbit  has  the  form  shown  in  Fig.  42.  The  computation  shows  that 
it  has  two  loops  and.  therefore,  that  it  had  two  cusps  symmetrical  with 
respect  to  the  x-axis  before  it  arrived  at  the  eject  ional  form.  After  this 
orbit  passes  beyond  the  ejectional  form,  with  decreasing  values  of  (',  it 
acquires  a  loop  about  I— n,  which  intersects  the  x-axis  |>eri>cndicularly  in 
the  negative  direction  from  ft  and  which  has  a  double  point  on  the  x-axis 
between  1  —  M  and  n.. 

Consider  the  further  evolution  of  the  |>eriodic  orbit.  If  the  small  loop 
al  H  nit  1  —  n  should  again  pass  to  the  ejectional  form,  the  ejectional  orbit  would 
be  exactly  of  the  tyix-  of  that  from  which  the  loop  developed.  It  is  improba- 
ble that  such  an  additional  ejectional  orbit  exists  for  another  value  of  ('. 

Now  consider  the  possibility  of  that  part  of  the  orbit  which  crosses 
the  x-axis  perpendicularly  in  the  positive  direction  between  1— p  and  n 
passing  through  an  eject ional  form.  It  can  not  pass  to  an  ejectional  form 
with  n  because,  in  accordance  with  the  general  conclusions  respecting  the 
motion  near  a  ix>int  of  ejection,  the  branches  of  the  curve  near  n  would  lie 
in  the  positive  direction  from  it.  and  the  partial  loop  about  n  just  Ix-fore  tin- 
eject  ional  form  was  reached  would  be  described  in  the  retrograde  direction. 
Hut  this  branch  of  the  curve  could  evolve  to  an  ejectional  form  with  1  —  n, 
when  the  orbit  would  have  the  form  shown  in  Fig.  47.  With  decreasing 
values  of  C  this  orbit  acquires  an  additional  loop  about  1—  M,  which  is 
described  in  the  retrograde  direction  and  which  intersects  the  x-axis  JHT- 
l>endicularly  in  the  negative  direction  between  1  -M  and  n.  This  loop  can 
expand  and  take  an  ejectional  form  with  n,  then  acquire  a  loop  about  M. 
which  can  become  an  ejectional  form  with  1  -  M,  and  so  on,  being  an  ejecti<  >n.d 
form  first  with  one  of  the  finite  masses  and  then  with  the  other  in  a  never- 
ending  sequence.  The  ejections  from  1  -//  are  all  in  th<%  negative  direction. 


522  PERIODIC    ORBITS. 

and  from  M  they  are  all  in  the  positive  direction.  It  is  probable,  though  not 
certain,  that  this  is  qualitatively  the  course  of  evolution  of  the  direct  satellite 
orbit  from  which  the  start  was  made. 

Now  consider  the  direct  satellite  orbit  about  1— M  which  enlarges  in 
the  negative  direction  and  which  approaches  the  ejectional  form  from  the 
positive  direction.  The  ejectional  form  was  found  by  computation  and  is 
shown  in  Fig.  44.  With  decreasing  C  this  orbit  acquires  a  loop  about  1  —  M, 
which  may  pass  to  the  ejectional  form  with  M,  as  shown  in  Fig.  48.  The  other 
branch  which  crosses  the  z-axis  perpendicularly  may  pass  to  the  ejectional 
form  with  1  — M.  With  decreasing  values  of  C  this  orbit  acquires  a  small 
loop  about  1  —  n  which  never  again  passes  through  the  ejectional  form. 
But  the  loop  about  M  enlarges  and  becomes  an  ejectional  form  with  1  —  n 
with  the  ejection  in  the  negative  direction.  Then  follows  a  loop  which 
becomes  an  ejectional  form  with  M>  followed  by  a  loop  about  M  which  be- 
comes an  ejectional  form  with  1  —  M,  and  so  on,  first  with  one  finite  body  and 
then  with  the  other  in  a  never-ending  sequence.  The  ejections  from  1— /x 
are  in  the  negative  direction,  and  from  n  they  are  in  the  positive  direction. 

There  is  a  third  direct  satellite  orbit  whose  evolution  has  not  been 
traced.  Only  a  conjecture  can  be  made  in  regard  to  it,  and  that  conjecture 
is  that  it  acquires  cusps  and  then  loops,  probably  about  the  region  of  the 
equilateral  triangular  points. 

251.  On  the  Evolution  of  Retrograde  Periodic  Satellite  Orbits. — Con- 
sider the  retrograde  periodic  satellites  about  1— M.  There  are  three  such 
orbits,  only  one  of  which  is  real  for  large  values  of  C.  The  numerical  experi- 
ments which  were  made,  §238,  indicate  that  only  one  of  them  is  real  for  any 
value  of  C. 

For  large  values  of  C  the  retrograde  periodic  orbit  about  1—  M  is  small 
and  nearly  circular  in  form.  As  C  diminishes  the  orbit  increases  in  size 
and  departs  widely  from  a  circle.  Consider  the  question  of  its  passing 
through  an  ejectional  form  with  1  —  /*.  If  the  periodic  orbit  should  approach 
the  ejectional  form  by  shrinking  upon  1  — /z  from  the  positive  or  the  negative 
direction,  just  before  arriving  at  the  ejectional  form  it  would  make  a  partial 
loop  about  1  —  M  in  the  retrograde  direction,  and  just  after  passage  through 
the  ejectional  form  it  would  make  a  complete  loop  about  1  —  /*  in  the  positive 
direction.  But  it  was  seen  in  §250,  in  connection  with  a  consideration  of 
an  ejectional  orbit  in  the  problem  of  two  bodies  referred  to  rotating  axes, 
that  this  is  impossible.  Hence  the  retrograde  satellite  orbit  about  1—  n 
can  not  become  an  ejectional  orbit  with  1—  n,  at  least  until  after  it  has 
passed  through  an  ejectional  form  with  n- 

Now  consider  the  possibility  of  the  retrograde  satellite  orbit  about 
1  —  n  passing  through  a  collisional  form  with  M-  Since  it  intersects  the  z-axis 
between  1  —  /*  and  M  such  an  orbit  must  be  one  in  which  the  collision  is  in  the 
negative  direction,  in  which  the  ejection  is  in  the  positive  direction,  in  which 


«t    \-i -.1:1 <>KHII- 

the  partial  loop  just  before  collision  is  descril>ed  in  the  |>i»iti\c  direction,  and 
in  which  the  complete  loop  just  :ifter  collision  is  described  in  the  retrograde 
direction.  Tin-  i-  precisely  the  way  in  which  such  :i  limiting  form  can  be 
p:i^>ed.  and  the  periodic  orbit  pa-se-  through  this  form.  An  orbit  of  tin- 
type, with  the  roles  of  1  -  M  and  ^  interchanged,  was  computed  and  i-  -ho\\  n 

in  I  ig.  :•>'.» 

After  the  retrograde  |M-riodic  orltit  about  1  —  n  passes  through  an  ejec- 
tional  form  \\ith  ^.  i'  acquire-  a  retrograde  loop  about  n  which  cr..—e<  the 
./•-axis  in  the  positive  direction  between  1  —  n  and  p.  This  loop  enlarges  and 
passes  through  an  ejectional  form  with  1  —  M,  after  which  it  acquires  u  retro- 
urade  loop  about  1  -  p,  which,  in  turn,  enlarges  and  passes  through  an  ejcc- 
tional  form  with  M-  This  process  continues,  the  form  becoming  ejectional 
nr-t  with  one  finite  mass  and  then  with  the  other  in  a  never-ending  sequence 
In  all  of  these  orbits  the  parts  near  the  ejection  points  are  on  the  negative 
side  of  1—  n  or  the  {x>sitivc  side  of  n,  and  never  between  1—  j*  and  M-  '  I"' 
orbit-  of  these  series  \\hich  are  closed  orbits  of  ejection  with  1  —  M  are  a  part 
of  those  which  \\ere  shown  to  exist  in  Chapter  XV  for  sufficiently  small 
values  «f  M:  ;"'d  those  which  are  closed  orbits  of  ejection  with  M  are  the 
corresjwnding  orbits  for  the  other  finite  mass. 

( 'oiisider  first  the  orbits  of  the  type  under  consideration  which  are  orbits 
of  collision  and  ejection  with  1— M-  All  these  orbits  are  orbits  of  ejection 
in  the  negative  direction;  they  have  double  points  on  the  .r-uxis  in  the  posi- 
tive direction  from  n,  and  intersect  the  x-axis  ixT|>endicuIarly  only  in  tin- 
negative  direction  from  I— n.  They  are  therefore  only  those  orbits  of  §226 
which  are  characterized  by  ejection  in  the  negative  direction  and  by  even 
values  of  j;  those  characterized  by  odd  values  of  j  have  a  different  origin. 

The  orbits  of  the  type  under  consideration  which  are  orbits  of  ejection 
and  collision  with  n  also  intersect  the  x-axis  perpendicularly  only  on  the 
negative  side  of  1— M-  On  interchanging  the  rdles  of  1— M  and  n  in  §226, 
orbits  of  ejection  from  n  in  the  |X)sitive  direction  were  proved  to  exist  for 
1  -M  sufficiently  small.  Those  which  are  characterized  by  odd  values  of  j 
intersect  the  x-axis  |x-rpendicularly  on  the  negative  side  of  I-M.  They 
are  of  the  type  of  the  part  of  those  under  consideration  which  are  (.rl.il- 
of  collision  and  ejection  with  p. 

To  summarize:  The  retrograde  |H-riodic  satellite  orbits  about  1-ji, 
with  decreasing  values  of  (\  go  through  an  infinite  series  of  ejectional  forms 
with  1  — M.  the  ejections  all  being  in  the  negative  direction,  and  these  orbits 
are  those  of  the  orbits  treated  in  §22(3  which  are  ejected  from  I-/*  in  the 
negative  direction  and  which  are  characterized  by  even  values  of  j.  The 
retrograde  |x>riodic  satellite  orbits  also  go  through  an  infinite  series  of  ejec- 
tional forms  with  n.  the  ejections  all  being  in  the  positive  direction,  and  these 
orbits  are  those  which  can  be  shown  to  exist  by  the  methods  of  §226,  and 
which  are  characterized  by  ejection  in  the  positive  direction  from  n  and  l>\ 
odd  values  of  j.  There  are  similar  retrograde  periodic  satellite  orbits  alnmt 


524 


PERIODIC    ORBITS. 


/z,  and  they  go  through  a  similar  series  of  critical  ejectional  forms  with  both 
1  —  ju  and  M-  The  ejections  from  1  —  M  and  M  are  also  respectively  in  the  nega- 
tive and  positive  directions,  but  those  which  are  ejectional  forms  with  1  — M 
are  characterized  by  odd  values  of  j,  while  those  which  are  ejectional  forms 
with  //  are  characterized  by  even  values  of  j.  Therefore,  the  retrograde 
periodic  satellites  about  the  two  finite  bodies  together  pass  through  all  the 
ejectional  forms  from  both  finite  bodies,  of  the  type  treated  in  §226,  in 
which  the  ejection  from  one  body  is  in  the  opposite  direction  from  the  other. 

252.  On  the  Evolution  of  Periodic  Orbits  of  Superior  Planets. — It  was 
shown  in  Chapter  XII  that  for  large  values  of  C  there  are  two  periodic 
orbits  in  which  the  infinitesimal  body  makes  simple  circuits  about  both  of  the 
finite  bodies  in  the  retrograde  direction.  When  the  system  is  referred  to 
fixed  axes,  one  of  the  orbits  is  direct  and  one  is  retrograde.  There  are  also 
four  orbits  in  which  the  coordinates  are  complex,  two  of  them  being  direct 
when  referred  to  fixed  axes  and  two  being  retrograde.  It  is  not  known 
whether  or  not  either  pair  of  the  complex  orbits  becomes  real  with  decreasing 
values  of  C.  Since  none  of  these  orbits  was  computed,  very  little  is  positively 
known  about  their  geometrical  characteristics  or  about  their  evolution. 

It  was  shown  in  §248  that  there  are  two  infinite  sets  of  orbits  which  are 
orbits  of  ejection  from  one  finite  body  and  of  collision  with  the  other.  One 
set  is  characterized  by  the  fact  that  the  ejection  from  each  finite  body  is  in 
the  direction  away  from  the  other.  Reasons  were  given  in  §249  for  believing 
that  they  are  limiting  forms  of  the  analytic  continuations  of  the  oscillating 
satellites  about  the  equilibrium  point  b.  The  other  set  is  characterized  by 
the  fact  that  the  ejection  from  each  finite  body  is  toward  the  other  finite 
body.  These  orbits  are  probably  limiting  forms  of  the  analytic  continua- 
tions of  retrograde  periodic  planetary  orbits.  The  probable  series  of  changes 
to  the  first  limiting  form  is  shown  qualitatively  in  Figs.  49  and  50.  Beyond 
the  limiting  form  the  orbits  acquire  loops  about  each  of  the  finite  bodies. 

There  are  two  retrograde  periodic  orbits  of  the  types  of  superior  planets. 
Probably  they  both  undergo  evolutions  to  limiting  forms  of  the  types  de- 
scribed. This  conjecture  is  supported  by  the  fact  that  two  closed  orbits  of 
ejection  were  found  by  computation,  §247,  for  a  related  type  of  orbits. 


Fio. 


FIG.  50. 


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